from the probability distribution, find the mean and standard deviation for the random variable x, which…

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.\n|x|p(x)|\n|0|0.145|\n|1|0.408|\n|2|0.264|\n|3|0.100|\n|4|0.083|\n\nμ = 1.866; σ = 1.111\nμ = 1.568; σ = 1.893\nμ = 1.568; σ = 1.111\nnone of these

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.\n|x|p(x)|\n|0|0.145|\n|1|0.408|\n|2|0.264|\n|3|0.100|\n|4|0.083|\n\nμ = 1.866; σ = 1.111\nμ = 1.568; σ = 1.893\nμ = 1.568; σ = 1.111\nnone of these

Answer

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:

$\mu = 1.568;\sigma = 1.111$