can you please help me out with these questions of mathematical Statistics

Questions 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 NV = 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 -, a y , 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(X3) 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 ( ) and Y ~ Exponential (3) and X and Y are indepen dent. Find Pr (X + Y