solve the following questions. These are stats questions that needs to be done by hand so dont use excel or any software for these but use basic formulas for all 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 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+y), 0

a){0.20.8 c)Y <0.5. 4) With reference to the joint probability density f(x1,x2)= { 2/3(x1+2x2) , 0

a) Find the moment generating function of z b) Identify the distribution of Z by recognizing the form of the moment generating function.

Hint: 12t / 2 e^-(1-2t) z/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(X) by differentiating the moment generating function.