Assist with the following questions:

47. Normal Distribution: Throughout these problems, let of and + be the pdf and edf, respectively, of the standard normal distribution Suppose Z is a standard normal random variable and let X = 32 + 1. (a) Express P(X 110). 50. More Transforming Normal Distributions (a) Suppose Z is a standard normal random variable and let Y = a7 +6, where a > 0 and 5 are constants. Show Y ~ N(b, a). (b) Suppose Y ~ N(, 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(, o')). 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. Compute43. 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. Practice Exam 1: All Questions, Spring 2014 (1) F(X) 1- 0.5 1.5- (iv) F(x) (v) F(I) (vi) F(x) 1. 0.5 - 0.5 - 0.5 - (vil) F(:) (vill) 0.5 - 44. Suppose X has range [0,1] and has edf F(x)=r brosrs1. Compute P( 4. Determine the pmf of X