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Chapter 4: Q.4.18 (page 171)

Let Xbe a Poisson random variable with parameter λ. What value of λmaximizes P{X=k},k≥0?

Short Answer

Expert verified

The solution isλ=k.

Step by step solution

01

Step 1: Given information

Let X be a Poisson random variable with parameter λ.

02

Step 2:Explanation

We have that

P(X=k)=λkk!e-λ

We are required to find λ>0such that for thatλfunction λ↦λkk!e-λmaximizes. Since 1/k!is a constant, we can move it out of our consideration. Also, since the logarithm is strictly increasing function, it is enough to find the maximum of following function

g(λ):=logk!P(X=k)=logλke-λ=klogλ-λ

Let's find stationary points.

We have that

dgdλ=kλ-1=0⇒λ=k

Since λ=kis the only stationary point, we have that for that λdensity function P(X=k)maximizes.

03

Step 3:Final answer

The solution is λ=k.

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