The probability distribution of a random variable X is given below
$X = x_{i}$ $P (X = x_{i})$
$1$ k
$2$ $2 k$
$3$ $3 k$
$4$ $4 k$
$5$ $5 k$
Find the value of k and the mean and variance of X.

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Solution

As we know that
$\sum P_{i} = 1$
$∴ k + 2 k + 3 k + 4 k + 5 k = 1$
$\Rightarrow k = \frac{1}{15}$
Thus the value of $k$ is $\frac{1}{15}$
$X = x_{i}$ $P (X = x_{i})$ $X P$ $X^{2} P$
$1$ $k$ $k$ $k$
$2$ $2 k$ $4 k$ $8 k$
$3$ $3 k$ $9 k$ $27 k$
$4$ $4 k$ $16 k$ $64 k$
$5$ $5 k$ $25 k$ $125 k$
$\sum X P = 55 k = 55 \times \frac{1}{15} = 3.66$ $\sum X^{2} P = 225 k 225 \times \frac{1}{15} = 15$
Mean of the given data $(¯ x) = \sum X P = 3.66$
Variance of the given data $= \sum X^{2} P - {¯ x}^{2} = 15 - {(3.66)}^{2} = 15 - 13.4 = 1.6$
Hence the mean and variance of the given data is $3.66$ and $1.6$ respectively.

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