MATH 2131 3.00 MS2Assignment 4Total marks = 40Question 1: Suppose that X ? Binomial(n, p) and Y ?

MATH 2131 3.00 MS2Assignment 4Total marks = 40Question 1: Suppose that X ? Binomial(n, p) and Y ? Binomial(m, p)and that X and Y are independent.(a) (2 marks). Find the moment-generating function (mgf) of X.(b) (4 marks). Determine the distribution of Z = n ? X by finding themgf of Z.(c) (3 marks). Use mgf’s to show that X + Y ? Binomial(n + m, p).Question 2: (3 marks). Let X1 , X2 , . . . be an infinite sequence of discreter.v.’s such thatx = ±n,1/2n,pXn (x) = 1 ? 1/n, x = 0,0,otherwise.Let Y be a degenerate r.v. where P (Y = 0) = 1. Show that the sequenceX1 , X2 , . . . converges in probability to Y .Question 3: (5 marks). Let X1 , X2 , . . . be i.i.d. Uniform[0,1] r.v’s. LetYn = max(X1 , X2 , . . . , Xn ). Show that the infinite sequence of r.v.’s Y1 , Y2 , . . .converges in probability to the degenerate r.v. Y where P (Y = 1) = 1.Question 4: (6 marks). A portable radio uses a type of battery that, afterinstallation, has a lifetime with a mean of 8 hours and a standard deviationof 8 hours. The radio is tested by running it continuously for 30 days withbatteries being replaced immediately after they run down. Approximate theprobability that 100 batteries will run the radio continuously for at least 30days.Question 5: (3 marks). Let X1 , X2 , . . . be i.i.d. r.v.’s each having the pdf(3×2 , 0