# Let $X$ have a poisson distribution with mean $4$.Find $(i) \;P(X\leq$3$)\qquad[e^{-4} = 0.0183].$

Toolbox:
• A random variable $X$ is said to have a poisson distribution of the probability mass function of $X$ is
• $P(X=x)=\large\frac{e^{-\lambda }\lambda^x}{x!}$$\qquad (x=0,1,2........ for some \lambda > 0) • Constants of a poisson distribution : Mean=Variance=\lambda • A continuous random variable X is said to follow a normal distribution with parameter \mu and \sigma (or \mu and \sigma^2) if the probability density function is • f(x)=\large\frac{1}{\sigma \sqrt{2\pi}}$$e^{-\large\frac{1}{2}(\frac{x-\mu}{\sigma})^2};-\infty < x < \infty,-\infty< \mu <0$ and $\sigma > 0$
• $X\sim N(\mu,\sigma)$
• Constants of a normal distribution :
• Mean =$\mu$,variance =$\sigma^2$,standard deviation =$\sigma$
Step 1:
$X\sim P(4)$
$\therefore P(X=x)=\large\frac{e^{\Large -4}4^{\Large x}}{x!}$$\qquad x=0,1,2....$
Step 2:
$P(X\leq 3)=P(X=0)+P(X=1)+P(X=2)+P(X=3)$
$\qquad\qquad=e^{\Large -4}\big[\large\frac{4^0}{0!}+\frac{4^1}{1!}+\frac{4^2}{2!}+\frac{4^3}{3!}\big]$
$\qquad\qquad=0.0183\big[\large\frac{(1+4+8)3+32)}{3}\big]$
$\qquad\qquad=0.0061[71]$
$\qquad\qquad=0.4331$