Let X be a Poisson random variable with a parameter of 100 U

Let X be a Poisson random variable with a parameter of 100. Use Markov\'s inequality to find a bound on P(X > 120). Use Chebyshev\'s inequality to find a bound on P(X > 120). Using the fact that for X_i ~ Poisson(1), independent random variables, X_1 + X_2 +... + Xn ~ Poisson(n), use the central limit theorem to approximate the value of P(X > 120).

Solution

a) P [ X > 120 ] < = E( X ) / 120...

E(X) / 120 = 100 / 120 = 0.8333...
so, P [ X > 120 ] is <= 0.8333..

b) P [ X > 120 ] = P [ | X - 100 | > | 120-100| ] = P [ | X - E(X) | > 20 ] <= VAR ( X ) / [ (20)^2 ] = 100 / 400 = 0.25
So, P [ X > 120 ] < = 0.25....

C) P [ X > 120 ] = P [ (X - 100 ) / sqrt(100) > (120-100) / sqrt(100) ] = P [ Z > 2 ] = 1 - P [ Z <=2]

= 1 - 0.9772499 = 0.02275013