Answer following questions. (solve them by statistics formulas, don't use excel to solve these): 1) Let X and Y be independent normal random variables with means X and Y and variances ² X and ²Y , respectively. Use moment generating functions to show that X Y has a normal distribution with mean X Y and variance ²X + ²Y

2) The mean of a random sample of size n = 25 is used to estimate the mean of an infinite population that has a standard deviation of = 2.4. What can we assert about the probability that the error will be less than 1.2, if we use

a) Chebyshev's theorem

b) the central limit theorem? 3) A random sample of size n = 25 from a normal population has mean x = 45.3 and standard deviation s = 7.9. Does this information tend to support or refute the claim that the mean of the population is 40.5?

4) The density of the Student-t distribution with 1 degree of freedom is given by f(t) = 1 / (1 + t² )^-1, < t < Verify the value given for t0.05 for = 1 in a table of values of t. Hint: The integral function of 1/(1 + x²) is arctan(x).

5) Derive the density of 1 e ^-X when X has the exponential distribution with = 1 using

a) the distribution function method

b) the transformation method ( apply it's theorem)

6) Use the discrete convolution formula, Theorem 6.10, to obtain the probability distribution of X + Y when X and Y are independent and each has the discrete uniform distribution on {0, 1, 2}