Let X be a random variable with p.d.f 𝑓(𝑥) = {23 , 𝑤ℎ𝑒𝑛 𝑥 = 113 , 𝑤ℎ𝑒𝑛 𝑥 = 20, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒Find themoment generating function , mean and standard deviation

Let X be a random variable with p.d.f 𝑓(𝑥) = {23 , 𝑤ℎ𝑒𝑛 𝑥 = 113 , 𝑤ℎ𝑒𝑛 𝑥 = 20, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒Find themoment generating function , mean and standard deviation6. Find the m g f of the pdf 𝑓(𝑥) = 𝑞𝑥−1𝑝, 𝑥 = 1,2,3, . . .. and hence evaluate its mean andvariance. (given p+q =1)

Problem Solving (15 pts)1. A discrete random variable X has this probability distribution.a) Complete the table (10 pts)Compute:b) 𝜇 = 𝐸(𝑋) = ∑[𝑥𝑃(𝑥)] = ___________________ (2 points)c) 𝑉𝑎𝑟(𝑋) = ∑[(𝑥 − 𝜇)2𝑃(𝑥)] = ___________________ (2 points)d) Standard deviation. ___________________ (1 points)Note: Answers from b – d, round off to 2 decimal places. For the table take all decimal places.𝑥 0 1 2 3 4𝑃(𝑥) 0.12 0.25 0.18 0.35 0.10𝑥𝑃(𝑥)𝑥 − 𝜇(𝑥 − 𝜇)2(𝑥 − 𝜇)2𝑃(𝑥)

Consider the random variable X ∼ N (5, 9); that is, a normal random variable with mean µ = 5and standard deviation σ = 3.(a) Evaluate the following:P(X ≤ 2), P(X > 2), P(X ≤ 8), P(X > 8).(b) Find the points x such that:P(X ≤ x) = 0.10, P(X ≤ x) = 0.25, P(X ≤ x) = 0.52, P(X ≤ x) = 0.99.Page 2 Page 2 of 3

For a probability of a random variable x is given below :X: 1 2 4 5 6P: 0.15 0.25 0.2 0.3 0.1What is the Standrard deviation of x ?(a) 1.49(b) 1.56(c) 1.69(d) 1.72

X is a random variable with mean 8 and variance 16. A function F(X) is defined asF(X) = (2X - 12).Question 7What are the mean and standard deviation of F(X)?Mean 16, standard deviation 32Mean 4, standard deviation 8Mean 4, standard deviation 4Mean 4, standard deviation 16