Question

For the discrete probability distribution below find the mean, variance, and
standard deviation.
x
0
1
2
3
4
5
P(x)
0.15
0.20
0.35
0.22
0.07
0.01

Answer 1

Mean/expectation: multiply each $x$ by the corresponding probability $P(x)$, and sum them up.

$E(X) = 0\times0.15 + 1\times0.20 + 2\times0.35 + 3\times0.22 + 4\times0.07 + 5\times0.01 \\\\ \implies \boxed{E(X) = 1.89}$

Second moment: same as mean, except we use $x^2$ in place of $x$.

$E(X^2) = 0^2\times0.15 + 1^2\times0.20 + 2^2\times0.35 + 3^2\times0.22 + 4^2\times0.07 + 5^2\times0.01 \\\\ \implies E(X^2) = 4.95$

Variance: variance is defined by

$V(X) = E\bigg((X-E(X))^2\bigg) = E(X^2) - E(X)^2$

so

$V(X) = 4.95 - 1.89^2 \implies \boxed{V(X) = 3.06}$

Standard deviation: this is simply the square root of variance,

$\sqrt{V(X)} = \sqrt{3.06} \approx \boxed{1.75}$