can you please help me with mathematical Statistics

1. /12 marks) Let X be a binomially distributed random variable with parameters n and p. i. (12 marks) Show that Var(X) = np(1 - p). (See hint on p. 91 of notes). 2. /8 marks) Let X be a random variable having a negative binomial distribution with parameters k and p. Show that the probability mass function of X sums to 1 over its support. You may make use of the identities given in Exercise 4.1.4. 3. /9 marks) Let X be a hypergeometrically distributed random variable with parameters N = 10 and n = 4. i. /3 marks) Give the support of the random variable X in each of the following three cases: M = 2, M = 5, and M = 8. ii. /6 marks) Draw line-point graphs for the probability mass function and cumulative distribution function of X in the case M = 5. 4. /5 marks) Let X be a random variable having a Poisson distribution with parameter A. Without using a moment-generating function, show that E(X) = 1. 5. /12 marks/ Each day, Bozo decides randomly which colour of shirt to wear, between green, yellow, red, and blue. He chooses green with probability -, yellow with probability -, and red with probability We can assume that Bozo's choice each day is independent of all other days. Determine the following: i. /5 marks) In the next eight days, what is the probability that Bozo wears green five times, yellow one time, red one time, and blue one time? ii. /7 marks) In the next four days, what is the probability that Bozo wears green at least twice and never wears red? 6. /12 marks/ Consider a random variable X that has a continuous uniform distribution with parameters a and b. Find E(X*) and thus show that the continuous uniform distribution is symmetrical (hint: see p. 72 of notes). You may take the results of Exercise 4.2.1. as given and do not need to derive them again. 7. /7 marks/ Suppose that X ~ Exponential (a ) and Y ~ Exponential (2 ) and X and Y are indepen dent. Find Pr (X +Y