# Random Variables

30)Let X be a Poisson random variable with parameter (lambda). Show that P {X=i} increases monotonically and then decreases monotonically as i increases, reaching its maximum when i is the largest integer not exceeding (lambda).

Hint: Consider P{X=i}/P{X=i-1}.

37) Let X1, X2, ...., Xn be independent random variables, each having a uniform distribution function of M, Fm (.), is given by

Fm(x)=x^n, 0(less than or equal to)x(less than or equal to) 1

What is the probability density function of M?

40) Suppose that two teams are playing a series of games, each of which i s independently won by team A with probability p and by team B with probability 1-p. The winner of the series is the first team to win four games. Find the expected number of games that played, and evaluate this quantity when p=1/2.

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#### Solution Summary

The solution addresses random variables as follows - 30)Let X be a Poisson random variable with parameter (lambda). Show that P {X=i} increases monotonically and then decreases monotonically as i increases, reaching its maximum when i is the largest integer not exceeding (lambda).

Hint: Consider P{X=i}/P{X=i-1}.

37) Let X1, X2, ...., Xn be independent random variables, each having a uniform distribution function of M, Fm (.), is given by

Fm(x)=x^n, 0(less than or equal to)x(less than or equal to) 1

What is the probability density function of M?

40) Suppose that two teams are playing a series of games, each of which i s independently won by team A with probability p and by team B with probability 1-p. The winner of the series is the first team to win four games. Find the expected number of games that played, and evaluate this quantity when p=1/2.