Find the expected value, variance and standard deviation of a random variable whose $p . m . f$ is.
$X = x$ $1$ $2$ $3$
$p (X = x)$ $\frac{1}{5}$ $\frac{2}{5}$ $\frac{2}{5}$

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Solution

$X = x$ 1 2 3
$P (X = x)$ 0.2 0.4 0.4
$\to$ PMF (probability mass function)
Expected value $= E [X] = 0.2 \times 1 + 0.4 \times 2 + 0.4 \times 3$
$= 2.2$
Variance $= E [X^{2}] - (E [X])^{2}$
$E [X^{2}] = 0.2 \times 1^{2} + 0.4 \times 2^{2} + 0.4 \times 3^{2} = 5.4$
Variance $= 5.4 - (2.2)^{2} = 0.56$
Standard deviation $= \sqrt{v a r i a n c e} = \sqrt{0.56} = 0.748$

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$X = x$	$1$	$2$	$3$
$p (X = x)$	$\frac{1}{5}$	$\frac{2}{5}$	$\frac{2}{5}$

Find the expected value, variance and standard deviation of a random variable whose p.m.f is.X=x123p(X=x)152525

X=x 12 3 P(X=x) 0.2 0.4 0.4→ PMF (probability mass function)Expected value =E[X]=0.2×1+0.4×2+0.4×3 =2.2Variance =E[X2]−(E[X])2E[X2]=0.2×12+0.4×22+0.4×32=5.4Variance =5.4−(2.2)2=0.56Standard deviation =√variance=√0.56=0.748

Find the expected value, variance and standard deviation of a random variable whose $p . m . f$ is.
$X = x$ $1$ $2$ $3$
$p (X = x)$ $\frac{1}{5}$ $\frac{2}{5}$ $\frac{2}{5}$