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Standard Deviation about Mean for Continuous Frequency Distributions

A random vari...

Question

A random variable $X$ has the following p.d.f.
$X$ $0$ $1$ $2$ $3$ $4$ $5$ $6$ $7$
$P (X = x)$ $0$ $k$ $2 k$ $2 k$ $3 k$ $k^{2}$ $2 k^{2}$ $7 k^{2} + k$
The value of $k$ is

\frac{1}{8}

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\frac{1}{10}

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Solution

The correct option is B $\frac{1}{10}$
$x = 7 \sum x = 0 P (X) = 1$
$\Rightarrow P (0) + P (1) + P (2) . . . P (7) = 1$
$\Rightarrow 0 + k + 2 k + 2 k + 3 k + k^{2} + 2 k^{2} + 7 k^{2} + k = 1$
$\Rightarrow 10 k^{2} + 9 k - 1 = 0$
$\Rightarrow 10 k^{2} + 10 k - k - 1 = 0$
$\Rightarrow 10 k (k + 1) - (k + 1) = 0$
$\Rightarrow (k + 1) (10 k - 1) = 0$
$k = - 1$ or $k = \frac{1}{10}$ .
But $0 \leq P (x) \leq 1$
Hence $k = \frac{1}{10}$ .

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A random variable X has the following probability distribution.

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