A random variable X has the following probability distribution:

Find the value of C and also calculate the mean of the distribution.

$C=\frac{1}{10}$, Mean = $\frac{133}{50}$

The correct answer is Option (1) → $C=\frac{1}{10}$, Mean = $\frac{133}{50}$

We know that $Σp_i = 1$

$⇒C+2C+2C+3C + C^2+2C^2 + 7C^2+C=1$

$⇒10C^2+9C-1=0$

$⇒(10C-1)(C + 1) = 0$

$⇒C =\frac{1}{10}$ or $C = -1$ but probability cannot be negative

$⇒C =\frac{1}{10}$

Mean = $Σp_i{x_i}^2=C×0+2C×1+2C×2 + 3C×3+C^2×4 + 2C^2 × 5+ (7C^2 + C) × 6$

$=56C^2 + 21C = 56 ×\frac{1}{100}+ 21 ×\frac{1}{10}=\frac{266}{100}=\frac{133}{50}$