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Chapter 7: Problem 6
A fair die is rolled 10 times. Calculate the expected sum of the 10 rolls.
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An urn contains 30 balls, of which 10 are red and 8 are blue. From this urn, 12 balls are randomly withdrawn. Let \(X\) denote the number of red, and \(Y\) the number of blue, balls that are withdrawn. Find \(\operatorname{Cov}(X, Y)\) (a) by defining appropriate indicator (that is, Bernoulli) random variables \(X_{i}, Y_{j}\) such that \(X=\sum_{i=1}^{10} X_{i}, Y=\sum_{j=1}^{8} Y_{j}\) (b) by conditioning (on either \(X\) or \(Y\) ) to determine \(E[X Y]\).
Individuals 1 through \(n, n>1\), are to be recruited into a firm in the following manner. Individual 1 starts the firm and recruits individual 2. Individuals 1 . and 2 will then compete to recruit individual 3. Once individual 3 is recruited, individuals 1,2 , and 3 will compete to recruit individual 4 , and so on. Suppose that when individuals \(1,2, \ldots, i\) compete to recruit individual \(i+1\), each of them is equally likely to be the successful recruiter. (a) Find the expected number of the individuals \(1, \ldots, n\) that did not recruit. anyone else. (b) Derive an expression for the variance of the number of individuals who did not recruit anyone else and evaluate it for \(n=5\).
A certain region is inhabited by \(r\) distinct types of a certain kind of insect species, and each insect caught will, independently of the types of the previous catches, be of type \(i\) with probability $$ P_{i}, i=1, \ldots, r \quad \sum_{1}^{r} P_{i}=1 $$ (a) Compute the mean number of insects that are caught before the first type 1 catch. (b) Compute the mean number of types of insects that are caught before the first type 1 catch.
Consider \(n\) independent trials, the ith of which results in a success with probability \(P_{i}\) (a) Compute the expected number of successes in the \(n\) trials - call it \(\mu\). (b) For fixed value of \(\mu_{1}\) what choice of \(P_{1}, \ldots, P_{n}\) maximizes the variance of the number of successes? (c) What choice minimizes the variance?
It follows from Proposition \(5.1\) and the fact that the best linear predictor of \(Y\) with respect to \(X\) is \(\mu_{y}+\rho \frac{\sigma_{y}}{\sigma_{x}}\left(X-\mu_{x}\right)\) that if $$ E[Y \mid X]=a+b X $$ then $$ a=\mu_{y}-\rho \frac{\sigma_{y}}{\sigma_{x}} \mu_{x} \quad b=\rho \frac{\sigma_{y}}{\sigma_{x}} $$ (Why?) Verify this directly.
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