7 Let X be a discrete random variable with values in { 0 , 1 , 2 , . . . , n } and moment… 7 Let X be a discrete random variable with values in {0, 1, 2, . . . , n} and moment generating function g(t). Find, in terms of g(t),…

7 Let X be a discrete random variable with values in { 0 , 1 , 2 , . . . , n } and moment… 7 Let X be a discrete random variable with values in {0, 1, 2, . . . , n} and moment generating function g(t). Find, in terms of g(t),…

9 Let X and Y be random variables with values in { 1 , 2 , 3 , 4 , 5 , 6 } with distri- bution… 9 Let X and Y be random variables with values in {1, 2, 3, 4, 5, 6} with distri- bution functions pX and pY given by pX (j) …

10 Show that if then and h ( z ) = h (1) = 1 – / 1 – 4 p q z 2 2 qz , ( p/q, if p = q, 1 , if p =… 10 Show that if then and h(z) = h(1) =…

11 Show that if X is a random variable with mean µ and variance s 2 , and if X * = ( X – µ ) /s… 11 Show that if X is a random variable with mean µ and variance σ2, and if X∗ = (X − µ)/σ is the standardized version of X,…

1 Let Z 1 , Z 2 , . . . , Z N desc r i b e a bra n c h ing p r o cess in w h i c h ea c h pa r e n… 1 Let Z1, Z2, . . . , ZN describe a branching process…

2 Let Z 1 , Z 2 , . . . , Z N desc r i b e a bra n c h ing p r o cess in w h i c h ea c h pa r e n… 2 Let Z1, Z2, . . . , ZN describe a branching process…

3 In the chain letter problem (see Example 10.14) find your expected profit if (a) p 0 = 1 / 2, p… 3 In the chain letter problem (see Example 10.14) find your expected profit if (a) p0 = 1/2, p1 = 0, and p2 = 1/2. (b) p0 = 1/6, p1 = 1/2, and…

4 Let S N = X 1 + X 2 + · · · + X N , where the X i ’s are independent random variables with… 4 Let SN = X1 + X2 + · · · + XN , where the Xi’s are independent random variables with common distribution having generating function…

5 We have seen that if the generating function for the offspring of a single parent is f ( z ),… 5 We have seen that if the generating function for the offspring of a single parent is f (z), then the generating function for the number of offspring after two generations is given by…

6 Consider a queueing process such that in each minute either 1 or 0 customers arrive with… 6 Consider a queueing process such that in each minute either 1 or 0 customers arrive with probabilities p or q = 1 − p, respectively. (The number p is called the arrival rate.) When a customer…