I. (50%) 1.1. (10%) Let X have a distribution with p.d.f.
$f(x) = \frac{e^{-x}}{(1+e^{-x})^2}, \quad -\infty < x < \infty.$
What is the distribution of $Y = 1/(1+e^{-X})$?

1.2. (10%) A portfolio consists assets A and B. The standard deviation of asset return for A and B is 100 and 150, respectively. Assume that the standard deviation of portfolio return is 220. What is the correlation of asset return A and B?

1.3. (10%) A state highway patrol periodically samples vehicle speeds at various locations on a particular highway system into order to test vehicle speeds $H_o: \mu \leq 65$ vs $H_a: \mu > 65$. A sample of 36 vehicles will be selected and the population standard deviation is assumed to be 12 miles per hour. At 0.05 level of significance, what is the probability of making a Type II error when the actual mean speed is 66 miles per hour? ($Z_{0.05} = 1.645$)

1.4. (10%) The data consists of n independently selected pairs $(X_i, Y_i), i=1,2,\dots,n$. The linear regression line of Y on X is $\hat{Y} = 30 - 1.5X$. The linear regression line of X on Y is $\hat{X} = 12 - 0.54Y$. What is the absolute value of Pearson correlation coefficient between variables X and Y?

a. What is the variance of $\bar{D}$?

b. If the two-sample t test is incorrectly used to analyze paired data, what will happen to the t value if $\rho > 0$ (underestimated or overestimated)? Justify your answer.

II. (50%) 1. (15%) Suppose X, Y, and Z are independent Poisson random variables, each with mean 1. Find a. P(X+Y+Z=4) b. $E[(X+Y)^2]$ c. $P\left(X=1 \mid \frac{X+Y}{2} = 2\right)$

a. How large a random sample must be taken in order that $P(|\bar{X}_n - p| \leq 0.1) \geq 0.75$ when p=0.2?

b. How large a random sample must be taken in order that $E(|\bar{X}_n - p|^2 \leq 0.01)$ when p=0.2?

3. (15%) Suppose that a random sample $X_1, \ldots, X_n$ is drawn from a normal distribution for which the value of the mean $\mu$ is unknown and the standard deviation is 2, and the following simple hypotheses are to be tested: $$H_0: \mu = -1$$ $$H_1: \mu = 1$$ Determine the minimum value of $\alpha + \beta$ (type I error+ type II error) that can be attained for each of following values of the sample size n: a. n=1 b. n=16

a. Set out complete analysis of variance table.

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