The defendant was convicted by a jury of unlawful unnatural sexual intercourse with Sam,1 a child under the age of sixteen, G. L. c. 265, § 23; indecent assault and battery on David, a child under the age of fourteen, G. L. c. 265, § 13B; two counts of indecent assault and battery on Sam, a child under the age of fourteen; and contributing to the delinquency of a child, David, G. L. c. 119, § 63.2 He attacks the convictions in three respects: (1) that there were improprieties in the prosecutor’s opening statement and closing argument; (2) that an expert witness called by the Commonwealth impermissibly vouched for the credibility of the child witnesses; and (3) that it was error for the trial judge to admit in evidence certain statements made by the defendant to the police, at least in the absence of factual findings on the issue of alleged police coercion.
We conclude that there was no error with respect to the prosecutor’s opening statement, and nonprejudicial error in the closing argument. There was no error in the admission in evidence of the defendant’s statements to the police. The defendant’s challenge to the testimony of the Commonwealth’s expert witness plunges us again into the confusion surrounding the efforts of experts to “educate” the jury regarding the characteristics of child abuse victims. We believe that the testimony in this case constituted impermissible profiling in that it identified supposedly general characteristics of abused children that paralleled too closely those of the child victims in the case itself. Accordingly, it was error to admit that testimony. Nevertheless, given the other evidence, and particularly the defendant’s admissions, we conclude that the outcome would not have been different had the error not occurred. Accordingly, we affirm.
*491. Material facts. The jury could permissibly have found the following facts. Sam, age ten, lived with his mother, brother, and two sisters, as well as his mother’s boyfriend (not the defendant). The defendant was a friend of Sam’s mother. Sam had known the defendant for three years. During the year before the night in question, the defendant slept at Sam’s house regularly. When at the house, the defendant customarily played basketball or watched television with the children, and took them to movies or a restaurant, often carrying on these activities with Sam alone. There was evidence that Sam had learning difficulties, and that he had been required to repeat the first grade.
On more than one occasion, the defendant touched Sam’s penis, lowering Sam’s boxer shorts when he did so. At other times, the defendant put Sam’s penis in his mouth. Although the defendant requested that Sam touch his (the defendant’s) penis with his hands and mouth, and even offered him money to do so, Sam refused. The defendant made Sam promise not to tell anyone of these contacts, and when Sam so promised, gave him a video game console as a gift.
On April 3, 1998, David, also age ten and a friend of Sam, went to Sam’s house to spend the night. Prior thereto, David had never met the defendant. During the evening, David walked to a variety store with the defendant and one of Sam’s sisters. At the store, the defendant purchased chips and candy for David. While walking home, the defendant asked David, in the sister’s presence, whether he “liked jerking off.” Later, the defendant told David that he would buy him everything he had bought for Sam, and a bicycle as well. Subsequently, the defendant gave Sam and David a sip of beer from a glass bottle, and asked David, outside of Sam’s presence, to show him the size of his penis.
Sam and David went to sleep by themselves in a room ordinarily shared by Sam and his brother (who was not present that night). At 3:30 a.m., the defendant entered that room, brought David under a blanket, rubbed David’s upper leg, placed his hand underneath David’s night clothes, and touched the top of David’s penis. Frightened and upset, David started to cry, and ran first into the bathroom, and then into another room where Sam’s sisters slept. The defendant followed him there, telling him to return to Sam’s room.
*50At about 5:00 a.m., David awakened Sam, and the boys played video games. At about 6:30 a.m., the defendant returned to Sam’s room. Frightened, David told the defendant that he had vomited during the night, and had to go home to take a shower. He said that he would return at 8:00 a.m. and promised not to tell what the defendant had done. Upon arriving at his home, David reported the defendant’s behavior first to his mother, and then to his uncle. The police were called, and David subsequently described the incident to them. The police went immediately to Sam’s house, where they arrested the defendant.
After the arrest, Sam disclosed for the first time that he had also been abused by the defendant. He stated that he was afraid to tell because he thought that the defendant might harm him or that people might call him gay. He also testified that he feared that the Department of Social Services would become involved, and that he might be taken away from his mother.
At the police station following his arrest, the defendant, having been read his Miranda rights, agreed to speak to the officers.3 A tape recording of the defendant’s interrogation was played at trial, during which the defendant admitted touching Sam inappropriately approximately four times. In his statement to the police, the defendant denied touching David, but admitted that he had a conversation with David in which he asked him, “Why do you want to masturbate like that at an early age?” The defense at trial was essentially that the boys fabricated the sexual abuse4; that David incorrectly identified the defendant because of the darkness (thereby suggesting that the perpetrator was actually Sam’s mother’s boyfriend); and that his admissions to the police were the product of coercion.
2. Opening statement and closing argument. The defendant attacks the prosecutor’s opening statement, as well as her closing argument. Timely objections having been lodged as to both, we review for prejudicial error. See Commonwealth v. Santiago, 425 Mass. 491, 500 (1999).
With respect to the opening statement, the defendant contends *51that the prosecutor’s use of a narrative style had the effect of suggesting that she had personal knowledge of the facts of the case, and therefore constituted improper vouching. “A prosecutor may not express a personal opinion as to the credibility of a witness or assert personal knowledge of the facts in issue.” Commonwealth v. Francis, 432 Mass. 353, 357 (2000). She may, however, state what she expects to be able to prove. Ibid. Here, the prosecutor did just that. There is nothing wrong with a narrative as long as it remains clear to the jury that the narrative is a prediction of what will be established by the evidence. Contrary to the defendant’s contention, Commonwealth v. Griffith, 45 Mass. App. Ct. 784, 785-786 (1998), where the prosecutor described evidence he was not in a position to introduce, is not helpful to the defendant given that, in the present case, the substance of the prosecutor’s opening was in fact borne out by the evidence — as the prosecutor would have reasonably expected it to be. Any possible prejudice was cured by the judge’s instruction that opening statements are not evidence. Commonwealth v. Simpson, 434 Mass. 570, 584 (2001).
The defendant launches multiple attacks on the prosecutor’s closing argument. We address each in turn. He objects first to the prosecutor’s assigning to him of “thoughts” regarding his molestation of the children, arguing that there was neither evidence of his thoughts nor other evidence from which inferences regarding his thoughts could reasonably be drawn. At the same time, he objects to the prosecutor’s use on several occasions of the term “cocky” as a characterization of the defendant. We agree that a prosecutor may neither engage in prejudicial characterization of the evidence nor argue conclusions that the evidence does not logically support. See Commonwealth v. Rivera, 52 Mass. App. Ct. 321, 326 n.3 (2001). Reasonable inferences from the evidence are obviously permissible. See Commonwealth v. Kater, 432 Mass. 404, 422 & n.14 (2000). Here, a reading of the argument as a whole convinces us that the prosecutor urged on the jury reasonable conclusions regarding the defendant’s motivations in dealing with the children, and while the tone was somewhat overwrought in places, the propositions were not unfair. Likewise, as the trial judge observed, describing the defendant as “cocky” was a comment *52both on his demeanor as a witness and on the plausibility of his defense. The jury could properly consider this aspect with other factors in order to arrive at a conclusion regarding the defendant’s credibility. The prosecutor’s comments did not suggest that she possessed knowledge of incriminating facts not contained in the evidence. See Commonwealth v. Valliere, 366 Mass. 479, 494 (1974).
The defendant challenges the prosecutor’s statements that “[the defendant] thinks he can walk into this courtroom and tell his preposterous story, with all of its inconsistencies, and lie to you and that you’ll buy it”; that the defendant assumed the child victims would be too “ashamed” and “scared” to testify; and that in fact the defendant ought to be the one who should be ashamed and scared. The defendant characterizes the prosecutor’s comments as an invitation to the jury to view the defendant unfavorably for his decision to exercise his constitutional rights to contest the charges, have a trial by jury, and confront the witnesses against him. We fail to see any improper reflection on the exercise of the defendant’s rights. The defendant testified, thereby subjecting that testimony to fair comment as to its credibility. “A prosecutor can address, in a closing argument, a witness’s demeanor, motive for testifying, and believability, provided such remarks are based on the evidence, or fair inferences drawn from it, and are not based on the prosecutor’s personal beliefs. . . . When credibility is an issue ... ‘it is certainly proper for counsel to argue from the evidence why a witness should be believed.’ ” Commonwealth v. Freeman, 430 Mass. 111, 118-119 (1999), quoting from Commonwealth v. Raymond, 424 Mass. 382, 391 (1997). Evidence-based comment that a witness, including the defendant, has been untruthful is equally permissible. Here, the prosecutor confined herself to acceptable comments regarding the defendant’s credibility. Nothing threatened the defendant’s rights. Contrast Commonwealth v. Person, 400 Mass. 136, 139-140 (1987); Commonwealth v. Kowalski, 33 Mass. App. Ct. 49, 53-54 (1992); Commonwealth v. Jones, 45 Mass. App. Ct. 254, 256-257 (1998).
Similarly, we detect no impermissible vouching in the prosecutor’s argument as to why the testimony of the child *53victims should be believed. It was permissible to suggest that the children had no motive to fabricate. See Commonwealth v. Krepon, 32 Mass. App. Ct. 945, 946-947 (1992). In a dispute that turned, in part, on the credibility of conflicting versions of events, the prosecutor was entitled to argue from the evidence why the Commonwealth’s witnesses should be believed. The argument did not have the “dogmatic ‘take it from me’ quality that the cases have declared to be impermissible.” Commonwealth v. Griffith, 45 Mass. App. Ct. at 787. None of the prosecutor’s remarks in context suggested that she had personal knowledge, or a personal belief, that the children were telling the truth. See Commonwealth v. Mitchell, 428 Mass. 852, 857 (1999).
The prosecutor’s references to “how hard” it must have been for Sam and David, and for David’s mother, to testify, as well as her comment “how awful as a parent this whole thing was,” crept perilously close to the line separating a proper discussion of the evidence from an improper appeal to jury sympathy. See, e.g., Commonwealth v. Harris, 11 Mass. App. Ct. 165, 176 (1981); Commonwealth v. Lorette, 37 Mass. App. Ct. 736, 742 (1994), S.C. 422 Mass. 1014 (1996). The references were perhaps defensible as factors bearing on the witnesses’ credibility. It was within the discretion of the trial judge to permit the statements, particularly in the light of instructions that were adequate to dispel any sympathy or prejudice the jury may have experienced. See Commonwealth v. Pearce, 427 Mass. 642, 645 (1998).
The same cannot be said with respect to the prosecutor’s closing statement, “Now is the time for justice. Find him guilty.” This kind of rhetorical flourish creates too great a risk of misinterpretation by the jury that there is a duty to convict, see Commonwealth v. Adams, 434 Mass. 805, 822 (2001), or that proper performance of their function requires a guilty verdict, see Commonwealth v. Degro, 432 Mass. 319, 328-329 (2000) . However, the statement was a brief error at the conclusion of a closing argument that spanned eleven pages. We do not approve of the comment, but we find it was not prejudicial.
Finally, it was not good practice for the prosecutor to discuss the definition of “reasonable doubt” in the closing argument. *54See Commonwealth v. Snow, 30 Mass. App. Ct. 443, 447 (1991). However, the definition she offered did not differ materially from that of Commonwealth v. Webster, 5 Cush. 295, 320 (1850). Furthermore, the judge correctly instructed on the subject, and also cautioned the jury that they must take instructions on the law exclusively from her. Accordingly, there was no prejudice. See Commonwealth v. Amico, 15 Mass. App. Ct. 650, 652 (1983).5
3. Expert testimony. The defendant challenges expert testimony offered by the Commonwealth on the subject of certain characteristics of sexually abused children, as well as the absence of an expert witness instruction. The defendant did not object during the testimony of the expert. At the close of that testimony, he objected to the entirety of it on the ground asserted here, and sought a mistrial. In light of our treatment of the subject, we assume that the question was preserved and therefore, the prejudicial error standard applies. See Commonwealth v. Loguidice, 420 Mass. 453, 456 (1995). With respect to the absence of an expert witness instruction, there was no objection, and we therefore determine whether there was error and, if so, whether such error created a substantial risk of a miscarriage of justice. See Commonwealth v. Colon, 49 Mass. App. Ct. 289, 291 (2000).6
It is within the trial judge’s discretion, subject to proper limiting instructions, to admit expert testimony on the general behavioral characteristics of sexually abused children. See Commonwealth v. Dockham, 405 Mass. 618, 627-630 (1989). The proffered rationale is that absent counseling, with respect to such characteristics, juries might unfairly impose standards of normalcy on child victims who, because of their experiences, were likely to act abnormally. “[T]he routine indicia of witness reliability — consistency, willingness to aid the prosecution, *55straightforward rendition of the facts — may, for good reason, be lacking.” Id. at 629-630.
In the aftermath of Dockham, certain general principles governing such testimony emerged. “The line between permissible and impermissible opinion testimony in child sexual abuse cases is not easily drawn.” Commonwealth v. Richardson, 423 Mass. 180, 186 (1996). While testimony regarding general behavioral characteristics of abused children is permitted, a linking of the generalized opinion to the experience of the actual child witness is impermissible vouching. Commonwealth v. Trowbridge, 419 Mass. 750, 759-760 (1995). If that link is avoided, and the testimony does not focus on the specific child witness, the testimony is usually allowed. Commonwealth v. Hudson, 417 Mass. 536, 541-543 (1994). However, the cases recognize that there will be instances where, even absent explicit linking of the opinion to the child witness, the jury may impermissibly be invited to find an identity between the hypothetical subjects of the expert testimony and the specific child witness who has testified before them. See Commonwealth v. Montanino, 409 Mass. 500, 504-505 (1991). “Even if [testimony pertaining to patterns of disclosure] falls short of a direct comparison between the specific complainant before the jury and the credibility of similarly situated children, it is impermissible if the opinion would inevitably be applied to the child alleging sexual abuse.” Commonwealth v. Colon, supra at 292-293.
However, these principles, while seemingly clear in the abstract, become hard to apply in actual practice. We confess to some difficulty in separating permissible “general” expert testimony from impermissible vouching. We understand that the jury’s fact-finding role is jeopardized where the expert’s description of typical characteristics matches the characteristics of the specific victims upon whom the jury’s attention is focused. But if there is not at least some relationship between the general characteristics and the actual case being tried, then there is presumably no justification for the admission of the “general” expert testimony in the first place. So the question remains how to distinguish in a given case between that expert testimony that fulfils a legitimate educational function helpful to the fact *56finder, and that expert testimony that unlawfully coaches the fact finder whether to accept or reject the testimony of particular witnesses.
In Commonwealth v. Federico, 425 Mass. 844, 850 n. 11 (1997), the court concluded that expert testimony regarding “common clinical phenomena seen in children who have been sexually abused[,] such as secrecy, delayed disclosure and retraction . . . [and] common behaviors seen in children who have been sexually abused which may appear strange or bizarre to a lay person and even inconsistent with a claim of sexual abuse” is admissible. However, the court decided that the actual hypothetical questions answered by the expert witness did not assume “common clinical phenomena,” but instead “assumed specific facts admitted in evidence through the testimony of the two child complainants.” Ibid. The court decided that “there [was] a significant risk that the jury could have understood [the expert’s] testimony to mean that sexual abuse had in fact occurred in this case. That is impermissible. See Commonwealth v. Colin C., [419 Mass. 54,] 60-61 [1994]. Because the only evidence against the defendant were the allegations of the complainants, at the very least [the expert’s] testimony tended to support their allegations and to buttress their credibility.” Id. at 851.
While we may never be able to define the dividing line precisely, we would uphold as permissible that expert testimony that explains to the jury that child abuse victims may behave in ways that to lay persons may seem illogical. The jury may then factor that advice into its deliberation, while still preserving its role as the ultimate decision maker with respect to child witnesses’ credibility. On the other hand, expert testimony that describes what a typical victim looks or acts like, and that suggests that child victims in a particular case have acted typically when compared to a “norm” of child victims, may not be admitted.
We recognize the apparent oddity that, under this approach, the more numerous the similarities between the individual child victim and a “standard” child victim created by an expert witness, the more likely it is that the expert testimony will be excluded. That is because our concern is not with relevance; the *57expert’s testimony regarding characteristics of child victims of sexual abuse was plainly “relevant.” The concern that we address is the fear that the jury’s responsibility ultimately to decide for itself whether the children are telling the truth will give way to expert opinion that paints the child witness as a victim. See Commonwealth v. Federico, supra at 851. Furthermore, we discern a significant difference between permissible expert testimony that suggests to the jury that certain behavior on the part of the victim, e.g., late reporting, does not necessarily exculpate the defendant, and impermissible expert testimony that other facts, e.g., the giving of gifts to the victim by the alleged perpetrator, may be a basis for finding the defendant guilty because they have occurred in other child abuse cases. We acknowledge that the distinction may be subtle in given situations, but we believe that the underlying principles are both sound and consistent with decisions since Dockham.7
We are of the opinion in the present case that the dividing line was crossed, and the expert vouched for the credibility of the complainants. The witness got off to what we consider a dubious start by announcing that he would testify on the “dynamics that exist in households where there is child sexual abuse.” This signaled that a profile was about to emerge. It is a fundamental principle of our system of criminal justice that we do not convict people of crimes on the basis of statistics or models; it appears equally that we do not attempt to determine whether someone has been a victim of crime on the same basis.
The witness then proceeded to identify not less than eight general characteristics of child abuse victims. Each characteristic was present in the evidence regarding either one or both of the two actual complainants. A brief description may be instructive. (1) Child abuse victims often are affected by cognitive limitations or learning difficulties, thereby rendering them more vulnerable to predators. Here, Sam had repeated the first grade. (2) The victims are often told by the perpetrator to keep the *58activity secret. Sam testified that the defendant said that he should not tell. (3) Victims often receive positive rewards for the sexual activity and for keeping the secret. Here, the defendant gave Sam a video game console and promised David that he would buy him things as well. (4) The child often does not disclose the abuse because he fears that someone will get hurt. Sam testified that he was afraid that the defendant would harm him. (5) Victims are especially vulnerable to trusted persons. In the present case, the defendant was a friend of Sam’s mother who was at Sam’s home regularly. (6) The victim is often drawn into the relationship by special favors, as here where the defendant took Sam to movies and restaurants. (7) The victim often fears that the fact of the sexual contact means that he is homosexual. Sam testified that another reason for not telling of the abuse was that he feared people would think he was gay. (8) There is a greater likelihood of immediate disclosure when there is no pre-existing relationship between abuser and victim. David had not known the defendant prior to the night that he stayed with Sam, and David disclosed the sexual contact the following day.
The vivid portrait of the child who does not disclose was, in essence, a portrait of Sam. Consistently, David, who did disclose immediately, did not share most of the enumerated characteristics. The invitation to the jury to judge these victims’ experiences on the basis of characteristics of victims in general was very great. It was not mitigated by the testimony regarding certain characteristics that were not present, specifically, that the victim is often a member of a family with difficulties in communication, and that female victims often have behavioral reactions that differ from those of male victims (wholly irrelevant in this case given that both victims were male). Nor is the testimony validated by the fact that the expert did not actually examine the victims and offered no testimony of actual events. While this may reduce the danger of implicit vouching in some cases, see Commonwealth v. Rather, 37 Mass. App. Ct. 140, 148 (1994), it does not dispose of the vouching concern in cases such as this where the profile so extensively matches the characteristics of the specific victims (albeit that the expert does not himself testify to the match).
*59It strikes us that what took place at this trial was far removed from what was contemplated when Commonwealth v. Dockham, supra, was decided. It is one thing to educate the jury to understand that child abuse victims may act in counterintuitive ways, and that excessive weight should not be given to factors such as failure to disclose when the child victim’s credibility is weighed. Cf. Commonwealth v. Quincy Q., 434 Mass. 859, 872-873 (2001) (expert testimony that absence of physical evidence of penetration does not exclude possibility that sexual abuse occurred); Commonwealth v. Colon, 49 Mass. App. Ct. at 292-293 (same). It is quite another to suggest to the jury that the events and feelings expressed by the child witnesses are the same as those experienced by other victims of abuse. That this has the effect of buttressing the witnesses’ credibility seems impossible to deny. See Commonwealth v. LaCaprucia, 41 Mass. App. Ct. 496, 500-502 (1996).
For the reasons stated, we conclude that it was error to admit the expert testimony in question. That, however, does not end the analysis, for we also decide that on this record the defendant was not prejudiced by the error. The defendant admitted to the police that he had touched Sam more than once, and a tape recording of that statement was played to the jury. See part 4, infra. Thereafter, the parties stipulated that “this type of touching had occurred with [Sam] approximately four times.” We view these admissions by the defendant as essentially rendering the expert testimony irrelevant.8 Likewise, the fact that no expert witness instruction was given, while perhaps harmful in another context, see Commonwealth v. Alphas, 430 Mass. 8, 13 (1999) (test is whether error is sufficiently significant in the context of the trial to make plausible an inference that result might have been different had it not been for the error), had little or no effect here given the defendant’s admissions and the concomitant reduction in importance of the expert testimony.
4. Voluntariness of admissions. The trial judge believed the *60police testimony that the defendant was given Miranda warnings, and that the defendant was not coerced or pressured to confess. That credibility determination is entitled to substantial deference. See Commonwealth v. Costa, 414 Mass. 618, 626 (1993); Commonwealth v. LeBlanc, 433 Mass. 549, 554 (2001). The defendant contends that the judge should have made subsidiary findings to support her ultimate finding that the admissions were voluntary. What the judge did was adequate. The reasons for the finding were quite evident. See Commonwealth v. Dayes, 49 Mass. App. Ct. 419, 420 (2000). That the judge did not make explicit subsidiary findings does not mandate reversal. See Commonwealth v. Brady, 380 Mass. 44, 52 (1980). It is also clear from her subsequent instruction to the jury that they also were required to decide, beyond a reasonable ■doubt, that the defendant’s statements were voluntary, see Commonwealth v. Harris, 371 Mass. 462, 469-470 (1976) (detailing the “humane practice” rule), that the judge understood the applicable standard of proof. See Commonwealth v. Mello, 420 Mass. 375, 382-383 (1995).
Judgments affirmed.
The names of the victims are pseudonyms.
The defendant was found not guilty of another indictment charging indecent assault and battery on David. An indictment charging him with attempting to commit a crime, G. L. c. 274, § 6, was filed without a change of plea with the consent of the defendant.
He subsequently asserted that his statements were involuntary, having been produced by threats by the police. See part 4, infra.
The defendant testified in this regard that the boys drank beer without his permission, and made up the allegations to avoid punishment.
We consider whether the cumulative effect of any errors requires a new trial, and conclude it does not.
Our analysis is not altered by the fact that the witness was not officially designated an “expert.” He testified on subject matter that is not normally within the knowledge of lay persons. His credentials suggested an advanced level of understanding with respect to such subject matter. The jury would have gotten the point that they were hearing from a witness who was performing a teaching function.
We observe that, in certain circumstances, child victims do not act counter-intuitively at all. We include, for example, the experience that a child often will not report the behavior of an authority figure who remains present in the child’s life. Under certain circumstances, the logic of that could be understandable by lay persons, and it is ultimately within the trial judge’s discretion whether to admit such testimony.
While the defendant did not admit to the rape, or to touching David, this is not material to the question of prejudice. This is because the improper testimony helped to establish the fact of Sam’s abuse, not the type, and the defendant’s admissions, standing alone, established the occurrence of the abuse.