Please solve the attached questions:

Determine the pmf of X. 43. For each of the following say whether it can be the graph of a cdf. If it can be, say whether the variable is discrete or continuous.33. Suppose that X ~ Bin(n, 0.5). Find the probability mass function of Y = 2X. 34. (a) Suppose that X is uniform on [0, 1]. Compute the pdf and edf of X. (b) If Y = 2X + 5, compute the pdf and cdf of Y. 35. (a) Suppose that X has probability density function fx(x) = Ae- for r 2 0. Compute the cdf, Fx (x). (b) If Y = X', compute the pdf and cdf of Y. 36. Suppose that X is a random variable that takes on values 0, 2 and 3 with probabilities 0.3, 0.1, 0.6 respectively. Let Y = 3(X -1)?. (a) What is the expectation of X?48. Transforming Normal Distributions Suppose Z ~ N(0,1) and Y = ez (a) Find the cdf Fy(a) and pdf fy(y) for Y. (For the CDF, the best you can do is write it in terms of + the standard normal cdf.) (b) We don't have a formula for $(=) so we don't have a formula for quantiles. So we have to write quantiles in terms of b-1. (i) Write the 0.33 quantile of Z in terms of b- (ii) Write the 0.9 quantile of Y in terms of $-1. (iii) Find the median of Y. 49. (Random variables derived from normal r.v.) Let X1, X2, ... Xn be i.i.d. N(0, 1) random variables. Let Yn = X/ + ...+ X (a) Use the formula Var(X,) = E(X?) - E(X;)' to show E(X?) = 1. (b) Set up an integral in r for computing E(X; ). For 3 extra credit points, use integration by parts show E(X) ) = 3. (If you don't do this, you can still use the result in part c.) (c) Deduce from parts (a) and (b) that Var( X}) = 2. (d) Use the Central Limit Theorem to approximate P(Ying > 110). 50. More Transforming Normal Distributions (a) Suppose Z is a standard normal random variable and let Y" = a7 + 6, where a > 0 and b are constants. Show Y ~ N(b, a). (b) Suppose Y ~ N(p, a). Show " follows a standard normal distribution. 51. (Sums of normal random variables) Let X be independent random variables where X ~ N(2,5) and Y ~ N(5,9) (we use the notation N(u, 5')). Let W = 3X - 27 +1. (a) Compute E(W) and Var(W). (b) It is known that the sum of independent normal distributions is normal. Compute