The theoretical (expected) frequencies are

$\begin{array}{ll}NP(0)\hfill & =(120)(0.22313)=27,\hfill \\ NP(1)\hfill & =(120)(0.334695)=40,\hfill \\ NP(2)\hfill & =(120)(0.25)=30,\hfill \\ NP(3)\hfill & =(120)(0.13)=16,\hfill \\ NP(4)\hfill & =(120)(0.05)=6,\hfill \\ NP(5)\hfill & =(120)(0.01)=1\hfill \end{array}$

i.e., 27, 40, 30, 16, 6, and 1.

* Example 4.73*: If

*X*and

*Y*are two independent random Poisson random variables such that Var[

*X*]+Var[

*Y*]=3. Find

*P*(

*X*+

*Y*<2)?

* Solution*: Let

*X*and

*Y*follow Poisson distribution.

Let *λ*_{1} be the mean of *X* and *λ*_{2} be the mean of *Y*.

Then by additive property *X*+*Y* follows Poisson distribution with mean *λ*_{1}+*λ*_{2}.

We have Var[*X*]+Var[*Y*]=*λ*_{1}+*λ*_{2}=3(given)

∴ *λ*=*λ*_{1}+*λ*_{2}

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