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*0.15 + 1*0.20 + 2*0.35 + 3*0.22 + 4*0.07 + 5*0.01 \\\\ \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*0.15 + 1^2*0.20 + 2^2*0.35 + 3^2*0.22 + 4^2*0.07 + 5^2*0.01 \\\\ \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,

`√(V(X)) = √(3.06) \approx \boxed{1.75}`