Consider the probability distribution shown here:

$\mathbf{p}\left(x\right)\mathbf{=}\frac{{\mathbf{3}}^{\mathbf{x}}{\mathbf{e}}^{\mathbf{-}\mathbf{3}}}{\mathbf{x}\mathbf{!}}\left(X=0,1,2.{.}^{}\right)$

a. Is x a discrete or continuous random variable? Explain.

b. What is the name of this probability distribution?

c. Graph the probability distribution for$\mathbf{x}\mathbf{\le }\mathbf{6}$ .

d. Find the mean and standard deviation of x.

The variable x follows a Poisson distribution.

The probability distribution is, $\mathrm{p}\left(\mathrm{x}\right)=\frac{3^{\mathrm{x}}{\mathrm{e}}^{-3}}{\mathrm{x}!}(\mathrm{X}=0,1,2.{.}^{})$

The Poisson probability distribution is a discrete probability distribution.

Here, x follows a Poisson distribution.

i.e;$\mathrm{x}~\mathrm{p}\left(\mathrm{\lambda }=3\right)$

The probability of an event can occur a countable number of times.

Therefore, the variable x is a discrete random variable.

The variable x is a discrete Poisson random variable.

Therefore,

The distribution is a Poisson probability distribution.

According to the Poisson probability distribution, the parameter $\lambda$is same as the Mean of the Poisson distribution.

The probability distribution is, $\mathrm{p}\left(\mathrm{x}\right)=\frac{3^{\mathrm{x}}{\mathrm{e}}^{-3}}{\mathrm{x}!}(\mathrm{X}=0,1,2..)$

Here, the parameter $\lambda$is3.

So,

The mean of the Poisson distribution is $\lambda =3$

The probability distribution table is calculated as

The graph of the probability distribution of$\mathrm{P}\left(\mathrm{x}\le 6\right)$ is given as follows:

Taking probabilities P(X=x) on Y-axis and x values on X-axis.

The mean of the Poisson distribution is,

$$\begin{gathered} \mathrm{Mean}={\mathrm{\mu }}_{\mathrm{x}}=\mathrm{\lambda } \\ =3 \end{gathered}$$

The parameter$\lambda$is the same as the mean of the Poisson distribution.

The standard deviation is calculated as:

$$\begin{gathered} \sigma =\sqrt{\lambda } \\ =\sqrt{3} \\ =1.732 \end{gathered}$$

Hence, the mean is 3, and the standard deviation is 1.732.