4. The number X of calls coming into Campus Safety follows a Poisson distribution with parameter 1 = 5. The probability that any particular call is a prank call is 0.2, independent of any other call....

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Extracted text: 4. The number X of calls coming into Campus Safety follows a Poisson distribution with parameter 1 = 5. The probability that any particular call is a prank call is 0.2, independent of any other call. Let Y be the number of prank calls received. This means that, given a value r for X, the variable Y is binomial with r trials and parameter 0.2. (a) Determine E(Y |X = x) and V(Y | X = x). (b) Use the previous part and the laws of total expectation and variance to find E(Y) and V(Y) (c) Show that Y also has a Poisson distribution. Find its parameter and verify your answer to the previous item that way.