7.75. The moment generating function of X is given byMX(t)=exp{2e 2} and that of Y by My(t)=(et+)¹0. If X and Y are independent, what are(a) P{X + Y = 2}?(b) P{XY = 0}?(c) E[XY]?

The moment generating function (MGF) of a random variable is:and the MGF of a random variable is:Also given that and are independent.If a random variable follows Poisson distribution with parameter then its MGF is given byBy using the uniqueness property of moment generating functions, we can write:Therefore, The probability mass function of the Poisson distribution with parameter is given by:If a random variable follows binomial distribution with parameter and then its MGF is given byBy using the uniqueness property of moment generating functions, we can write:, , and Therefore, The probability mass function of the binomial distribution with parameter and is given by:(a) The probability of is calculated as follows:Therefore, the probability of is .(b) Since and are independent, the probability is can be found using the fact that the product of and is only if either or is :So, the probability that is .(c) The expected value of the product for independent random variables and is given by:So, the expected value of is .(a) The probability of is .(b) The probability that is .(c) The expected value of is .