We have to find `E(X)` first:

`E(X)` `=8times1/8+12times1/6` `+16times3/8+20times1/4` `+24times1/12` `=16`

Then:

`V(X)` `=sum[{X-E(X)}^2*P(X)]`

`=(8-16)^2 times 1/8 + (12-16)^2 times 1/6 ` `+ (16-16)^2 times 3/8 + (20-16)^2 times 1/4 ` `+ (24-16)^2 times1/12`

`=20`

Checking this using the other formula:

V(X) = E(X 2) − [E(X)]2

For this, we need to work out the expected value of the squares of the random variable X.

X `8` `12` `16` `20` `24`
X2 `64` `144` `256` `400` `576`
P(X) `1/8` `1/6` `3/8` `1/4` `1/12`

`E(X^2)=sumX^2P(X)`

`=64times1/8+144times1/6+` `256times3/8+` `400times1/4+` `576times1/12`

`=276`

We found E(X) before: `E(X) = 16`

V(X) = E(X2) − [E(X)]2 = 276 − 162 = 20, as before.

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