Let be the probability of getting heads in a coin toss. Assuming that is a random variable, we want to estimate the probability distribution of from observed data of independent coin tosses (Bernoulli trials). Here, we introduce a prior distribution of () that follows a beta distribution , where and are the parameters. The probability density function of the beta distribution is given by
where is the beta function.
(1) Describe briefly what kind of distribution is.
(2) Given that heads appeared times in trials, determine the likelihood function for this observation .
(3) According to Bayes' theorem, the posterior distribution of after observing can be calculated as . Here, is a normalization constant that adjusts the integral of the posterior distribution to be 1. Given a prior distribution and observed data consisting of heads in trials, derive the posterior distribution of .
Let and be random variables representing scores in math and English, respectively. Assume and are independent and both normally distributed with mean 50 and standard deviation 10.
(1) Let be the total score and be the difference in scores . Calculate the variances of and : and .
(2) Show that and are uncorrelated.
(3) If two random variables follow a bivariate normal distribution and are uncorrelated, they are independent. Using this property, calculate the covariance of and given that is fixed at (where is a constant) : .
(4) Calculate the correlation coefficient of and given :
(5) Even when math and English scores are independent, a negative correlation emerges when selecting only examinees with high total scores. Briefly explain why, based on the results above.
When reporting experimental results, it is recommended to include effect sizes alongside -values and statistical significance. Cohen's (the difference between group means divided by within-group standard deviation) is a common effect size.
(1) Both the -statistic and Cohen's are standardized measures of mean differences between two groups, yet they provide different types of information. Explain the difference using the term 'sample size.'
(2) When is it particularly important to report effect sizes? Describe the circumstances and explain why.