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The theoretical (expected) frequencies are

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

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 λ1 be the mean of X and λ2 be the mean of Y.

Then by additive property X+Y follows Poisson distribution with mean λ1+λ2.

We have Var[X]+Var[Y]=λ1+λ2=3(given)

λ=λ1+λ2

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