Answer following questions. (solve them by statistics formulas, don't use excel to solve these): 1) In a certain city, the daily consumption of electric power (in millions of kilowatt-hours) can be treated as a random variable having a gamma distribution with = 3 and = 2. If the power plant of that city has a daily capacity of 12 million kilowatt-hours, what is the probability that this power supply will be insufficient on any given day? 2) Two independent binomially distributed random variables have each a success probability of 0.4 and 2 trials. a) Find the joint probability distribution. b) Find the probability that the second random variable is greater than the first. 3) Let two random variables have the joint density f(x, y) = { 6/ 5 (x + y² ), 0 < x < 1, 0 < y < 1 , 0 elsewhere. Find the probabilities that

a) {0.2 < X < 0.5} {0.4 < Y < 0.6}

b) X > 0.8, c) Y < 0.5 4) With reference to the joint probability density f(x1, x2) = { 2 /3 (x1 + 2x2), 0 < x1 < 1, 0 < x2 < 1, 0, elsewhere find an expression for the conditional density of the second random variable when the first takes on the value x1 5) With reference to question 3 above, find the expected value of the random variable whose values are given by g(x, y) = x ²y 6) Let Z have a normal distribution with = 0 and ² = 1 a) Find the moment generating function of Z² b) Identify the distribution of Z² by recognizing the form of the moment generating function Hint: 1 2t / 2 * e ^(12t)z²/2 is the density of a normal distribution with mean 0 and variance 1/(1 2t).

7) Let X have the probability density function f(x) = { ( 2e -^2x , for x > 0, 0, elsewhere. a) Find the moment generating function

b) Obtain E(X) and E(X²) by differentiating the moment generating function.