Question

Find the variance and standard deviation of the probability distribution of the random variable X, which can take only the values 3,5, and 7, given that P (3) = (7)/(30),P(5) = (1)/(3),P(7) = (13)/(30)

Answer 1

In order to find the variance ( $\sigma^2$ ) of the given distribution, we will use the following formula:

$\sigma^2=\sum x^2\times p(x)-\mu^2$

In this example:

$\mu=\frac{27}{5}$

Now we will find the sum:

$\sum x^2\times p(x) = 3^2\times\frac{7}{30}+5^2\times\frac{1}{3}+7^2\times\frac{13}{30} = \frac{95}{3}$

Putting all together we have:

$\sigma^2=\sum x^2\times p(x)-\mu^2 = \frac{95}{3} - (\frac{27}{5})^2 = \frac{188}{75}$

Standard deviation: $\sigma = \sqrt{\sigma^2} = \sqrt{\frac{188}{75}} = 1.5832$