Question

How do you find the mean of the random variable `x`?

X= 5,10,15,20,25
P(x) = 1/5, 1/5 , 1/5, 1/5, 1/5

What is the variance and standard deviation of the random variable `x`?
What is the standard deviation of the random variable `x`?

Answer 1

`"mean " E(X)=15`

`Var(X)=50`

`sd=7.07`

Explanation:

For a probability distribution

`color(red)(E(X)=sum_(all x)xP(X=x))---(1)`

in this case we have

`E(X)=5xx1/5+10xx1/5+15xx1/5+20xx1/5+25xx1/5`

`E(X)=1+2+3+4+5`

`E(X)=15`

the variance is calculated by

`color(blue)(Var(X)=E(X^2)-E^"(X))---(2)`

where `color(blue)(E(X^2)=sum_(allx)x^2P(X=x)--(3))`

`E(X^2)=5^2xx1/5+10^2xx1/5+15^2xx1/5+20^2xxx1/5+25^2xx1/5`

`E(X^")=1/5(25+100+225+400+625)`

`E(X^2)=275.`

`Var(X)=275-15^2=275-225=50`

`sd=sqrtVar(X)=sqrt50=7.07`