please help with practice problems and provide explanation

1. (a) Let Y1 and Y2 be independent random variables and let W = 1"1 + Y2. Let Y1 ~ Bin(5.i and w ~ Bin(20,). Show that Y2 ~ 13111053). (b) Let Y1, Y2, Y3 be a random sample from N(1, 1). Let 17 = % and 5'2 = izii'} 102. What is the distribution of U = 252. Using the distribution of U, nd 13(52) and wsz). 2. Let Y1, Y2, ..., Yn be a random sample from N(u, o2). Let Y = _ Er_, Y; and let $2 = n- 1Et= ( Y , - P) 2. a) Suppose that o = 9 and it is desired to have sample mean Y within 1.5 units of the population mean / with probability 0.95. What is the sample size n required to achieve this? b) Suppose that the variance o is unknown and it is desired to have the sample mean within 'S units of the population mean / with probability 0.9. What is the sample size n required to achieve this? 3. Let Y1, Y2, .., Yo be a random sample from N(0, 1) and let Y10 be another independent random variable from N(0, 1). Let Y = =>?_, Y, and U = >?_, (Y, - Y)2. What is the distribution of? a) Y; b) U; c) T = Zi1 Y?; d) V = - U+Y20 e) W = 494+Y10 U. Let Y1, Y2, , Yn be a random sample from a population having mean p and standard deviation 3. a]: Let n = 36. Using the central limit theorem, find an approximate value of PG? ul 5 0.5). b) Find the approximate sample size 11 required so that Pa? .u| s 1.2) = 0.99. 5. Let X ~ Bin(25, ). Find P(X 2 13) using the following methods: a) the normal approximation to binomial (use continuity correction); b) the exact binomial probabilities given at the end of the textbook. 6. (Level of difficulty for this problem may be higher than others) Let Y, and Y2 be random variables having the joint probability density function (8y172, Ogy1 1, Ogy2