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Probability: Moment Generating Functions and Poisson Process

1.) Let X be a discrete random variable with probability mass function
Pr {X=k} = c(1+ k^2) for k= -2, -1, 0, 1, 2.
a) Determine c.
b) Determine Pr {X <= 0}
c) Determine the mean of X
d) Why is the previous answer fairly obvious?
e) Determine the variance of X
f) Compute Pr {X=2 | X >= 0}
g) Determine the moment generating function of X

2.) Let Y be a Poisson random variable with parameter 1.5
a) Compute Pr {Y=0}
b) Compute Pr {Y <= 1}
c) Compute Pr {Y=0 |Y <= 1}
d) What is E[Y]?
e) Determine the variance of Y
f) Determine the moment generating function

3) Suppose the random variable S has moment generating function of
(q + p *(e^t))^n where q= 1- p, 0 < p < 1 and n is a positive integer. Find the mean and
variance of Y.

Solution Preview

1.a.)
because total probability = 1
=> sum(P[k] for k = -2,-1,0,1,2) = 1
=> P(-1) + P(-2) + P(0) + P(1) + P(2) = 1
=> c.(1+(-2)^2) + c.(1+(-1)^2) + c.(1+(0)^2) + c.(1+1^2) + c.(1+2^2)=1
=> c*{(1+4) + (1+1) + (1+0) + (1+1) + (1+4)} = 1
=> c*(5+2+1+2+5) = 1
=> c = 1/15 --Answer

b.)
P(X<=0) == P(X =-2) + P(X=-1) + P(X = 0)
=> P(X<=0) = (1/15)*{(1+(-2)^2) + (1+(-1)^2 + (1+0^2)}
=> P(X<=0) = (1/15)*{5+2+1) = 8/15 --Answer

c.)
mean of X = <X>
<X> = (for Xi =-2,-1,0,1,2)sum{Xi * P(Xi)}
=> <X> = {(-2)*P(-2) + (-1)*P(-1) + 0*P(0) + 1*P(1) + 2*P(2)}
=> <X> = {-2*(1+(-2)^2) - (1+(-1)^2) + 0 + (1+1^2) + ...

Solution Summary

Proabilities are determined and moment generating functions are calculated.

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