Yahir Haas

2022-01-28

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?

sineurosi0f

Expert

mean E(X)=15
Var(X)=50
sd=7.07
Explanation:
For a probability distribution
$E\left(X\right)=\sum _{allx}xP\left(X=x\right)$...(1)
in this case we have
$E\left(X\right)=5×\frac{1}{5}+10×\frac{1}{5}+15×\frac{1}{5}+20×\frac{1}{5}+25×\frac{1}{5}$
E(X)=1+2+3+4+5
E(X)=15
the variance is calculated by
$Var\left(X\right)=E\left({X}^{2}\right)-E\left(X\right)$...(2)
where $E\left({X}^{2}\right)=\sum _{allx}{x}^{2}P\left(X=x\right)$...(3)
$E\left({X}^{2}\right)={5}^{2}×\frac{1}{5}+{10}^{2}×\frac{1}{5}+{15}^{2}×\frac{1}{5}+{20}^{2}×x\frac{1}{5}+{25}^{2}×\frac{1}{5}$
$E\left(X\right)=\frac{1}{5}\left(25+100+225+400+625\right)$
$E\left({X}^{2}\right)=275.$
$Var\left(X\right)=275-{15}^{2}=275-225=50$
$sd=\sqrt{Var\left(X\right)}=\sqrt{50}=7.07$

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