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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…

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),…

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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…

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)  …

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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) = 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) = 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) =…

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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 s 2 , and if X * = ( X – µ ) /s…

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 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 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…

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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 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 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…

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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) p 0 = 1 / 2, p…

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…

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4 Let S N = X 1 + X 2 + · · · + X N , where the X i ’s are independent random variables with…

## 4 Let S N = X 1 + X 2 + · · · + X N , where the X i ’s are independent random variables with…

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 ),…

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…

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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…

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…

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