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Question

# If a random variable X follows a binomial distribution with mean 3 and variance 3/2, find P (X ≤ 5).

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Solution

## $\mathrm{Mean}\left(np\right)=3\mathrm{and}\mathrm{variance}\left(npq\right)=\frac{3}{2}\phantom{\rule{0ex}{0ex}}\therefore q=\frac{1}{2}\phantom{\rule{0ex}{0ex}}\text{and}p\mathit{}=1-\frac{1}{2}\phantom{\rule{0ex}{0ex}}n=\frac{\mathrm{Mean}}{p}\phantom{\rule{0ex}{0ex}}⇒n=6\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$ $\mathrm{Hence},\mathrm{the}\mathrm{distribution}\mathrm{is}\mathrm{given}\mathrm{by}\phantom{\rule{0ex}{0ex}}P\left(X=r\right)={}^{6}C_{r}{\left(\frac{1}{2}\right)}^{r}{\left(\frac{1}{2}\right)}^{6-r},r=0,1,2...6\phantom{\rule{0ex}{0ex}}=\frac{{}^{6}C_{r}}{{2}^{6}}\phantom{\rule{0ex}{0ex}}\therefore P\left(X\le 5\right)=1-P\left(X=6\right)\phantom{\rule{0ex}{0ex}}=1-\frac{1}{64}\phantom{\rule{0ex}{0ex}}=\frac{63}{64}$

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