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

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 $x$ $x$ ?
What is the standard deviation of the random variable $x$ $x$ ?

1 Answer

Jan 31, 2018

$mean E (X) = 15$ $"mean " E(X)=15$

$V a r (X) = 50$ $Var(X)=50$

$s d = 7.07$ $sd=7.07$

Explanation:

For a probability distribution

$E (X) = \sum a l l x x P (X = x) - - - (1)$ $color(red)(E(X)=sum_(all x)xP(X=x))---(1)$

in this case we have

$E (X) = 5 \times \frac{1}{5} + 10 \times \frac{1}{5} + 15 \times \frac{1}{5} + 20 \times \frac{1}{5} + 25 \times \frac{1}{5}$ $E(X)=5xx1/5+10xx1/5+15xx1/5+20xx1/5+25xx1/5$

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

$E (X) = 15$ $E(X)=15$

the variance is calculated by

$V a r (X) = E (X^{2}) - E (X) - - - (2)$ $color(blue)(Var(X)=E(X^2)-E^"(X))---(2)$

where $E (X^{2}) = \sum a l l x x^{2} P (X = x) - - (3)$ $color(blue)(E(X^2)=sum_(allx)x^2P(X=x)--(3))$

$E (X^{2}) = 5^{2} \times \frac{1}{5} + 10^{2} \times \frac{1}{5} + 15^{2} \times \frac{1}{5} + 20^{2} \times x \frac{1}{5} + 25^{2} \times \frac{1}{5}$ $E(X^2)=5^2xx1/5+10^2xx1/5+15^2xx1/5+20^2xxx1/5+25^2xx1/5$

$E (X) = \frac{1}{5} (25 + 100 + 225 + 400 + 625)$ $E(X^")=1/5(25+100+225+400+625)$

$E (X^{2}) = 275 .$ $E(X^2)=275.$

$V a r (X) = 275 - 15^{2} = 275 - 225 = 50$ $Var(X)=275-15^2=275-225=50$

$s d = \sqrt{V} a r (X) = \sqrt{50} = 7.07$ $sd=sqrtVar(X)=sqrt50=7.07$

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