Solutions - 5130-05

(a)

(1) Define Poisson approximation:

$Y\sim \text{Pois}(np)=\text{Pois}(1)$.

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(2) Estimate error:

$$|P_X(k)-P_Y(k)|\le n{p}^2=10{\left(\frac{1}{10}\right)}^2=\frac{1}{10}$$

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(b)

(1) Compute $P[X\le 1]$ using the binomial distribution:

$$P[X\le 1]=\left(\genfrac{}{}{0px}{}{10}{0}\right){\left(\frac{1}{10}\right)}^0{\left(\frac{9}{10}\right)}^{10}+\left(\genfrac{}{}{0px}{}{10}{1}\right){\left(\frac{1}{10}\right)}^1{\left(\frac{9}{10}\right)}^9\approx 0.7361$$

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(2) Compute $P[X\le 1]$ using the Poisson distribution:

$$P[X\le 1]=e^{-1}\frac{1^0}{0!}+e^{-1}\frac{1^1}{1!}\approx 0.7358$$

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(3) Compute error:

$$\text{error}=|0.7361-0.7358|=0.0003$$