Question

Find the standard deviation of the expected value of the probability distribution, X={1, 2, 3, 4, 5} P(X)={.10, .35, .15, .23, .17}

Answer 1

To find the standard deviation of the probability distribution, we can use the following formula:

$$\sigma=\sqrt{\sum(x_i-\mu)^2\cdot P(x_i)}$$

where:

$x_i:$ the $i^{th}$ value

$\mu:$ the mean of the distribution

$P(x_i):$ the probability of the $i^{th}$ value

$\mu=1\cdot0.10+2\cdot0.35+3\cdot0.15+4\cdot0.23+5\cdot0.17=3.02$ $X-\mu=\begin{Bmatrix} 1-3.02, 2-3.02, 3-3.02, 4-3.02, 5-3.02 \end{Bmatrix}=$

$=\begin{Bmatrix} -2.02, -1.02, -0.02, 0.98, 1.98 \end{Bmatrix}$

$\sigma=\sqrt{(-2.02)^2\cdot 0.10+(-1.02)^2\cdot 0.35+(-0.02)^2\cdot 0.15+0.98^2\cdot 0.23+1.98^2\cdot 0.17}=$

$=\sqrt{1.6596}=1.288$