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Chapter 4: Q.4.19 (page 171)

Show that Xis a Poisson random variable with parameter 位, then

EXn=位E(X+1)n-1

Now use this result to compute EX3.

Short Answer

Expert verified

EX3=位3+3位2+位

Step by step solution

01

Step 1: Given information

Given in the question thatXis a Poisson random variable with parameter 位, then E[Xn]=位E[(X+1)n鈭�1]. We need to find E[X3]

02

Step 2: Explanation

Using the theorem about the mean of function of random variable,

we have that

EXn=鈭�k=0鈭�kn位kk!e-位=鈭�k=1鈭�kn位kk!e-位=位鈭�k=1鈭�kn位k-1k!e-位

=位鈭�k=1鈭�kn-1位k-1(k-1)!e-位=位鈭�k=0鈭�(k+1)n-1位kk!e-位=位E(X+1)n-1

which had to be proved. Using that, we have that

EX3=位E(X+1)2=位EX2+2X+1=位EX2+2E(X)+1

Now, use that EX2=Var(X)+EX2=位+位2

and thatEX=位,

so we have that the expression above is equal to

=位位+位2+2位+1=位3+3位2+位

03

Step 3:Final answer

EX3=位3+3位2+位

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