Question

from the probability distribution, find the mean and standard deviation for the random variable x, which represents the number of cars per household in a town of 1000 households.

x	p(x)
0	0.145
1	0.408
2	0.264
3	0.100
4	0.083

μ = 1.866; σ = 1.111
μ = 1.568; σ = 1.893
μ = 1.568; σ = 1.111
none of these

Explanation:

Step1: Calculate the mean formula

The mean $\mu$ of a discrete - random variable is given by $\mu=\sum_{x}x\cdot P(x)$.
\[

$$\begin{align*} \mu&=(0\times0.145)+(1\times0.408)+(2\times0.264)+(3\times0.100)+(4\times0.083)\\ &=0 + 0.408+0.528 + 0.3+0.332\\ &=1.568 \end{align*}$$

\]

Step2: Calculate the variance formula

The variance $\sigma^{2}=\sum_{x}(x - \mu)^{2}\cdot P(x)$.
\[

$$\begin{align*} (0 - 1.568)^{2}\times0.145&=( - 1.568)^{2}\times0.145=2.458624\times0.145 = 0.3565\\ (1 - 1.568)^{2}\times0.408&=( - 0.568)^{2}\times0.408 = 0.322624\times0.408=0.1316\\ (2 - 1.568)^{2}\times0.264&=(0.432)^{2}\times0.264 = 0.186624\times0.264 = 0.0493\\ (3 - 1.568)^{2}\times0.100&=(1.432)^{2}\times0.100=2.050624\times0.100 = 0.2051\\ (4 - 1.568)^{2}\times0.083&=(2.432)^{2}\times0.083 = 5.914624\times0.083=0.4909 \end{align*}$$

\]
\[

$$\begin{align*} \sigma^{2}&=0.3565+0.1316 + 0.0493+0.2051+0.4909\\ &=1.2334 \end{align*}$$

\]

Step3: Calculate the standard - deviation formula

The standard deviation $\sigma=\sqrt{\sigma^{2}}$. So $\sigma=\sqrt{1.2334}\approx1.111$.

Answer:

$\mu = 1.568;\sigma = 1.111$