A random variable X has the following probability distribution:
|
$X$ |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
|
$P(X)$ |
$C$ |
$2C$ |
$2C$ |
$3C$ |
$C^2$ |
$2C^2$ |
$7C^2 + C$ |
Find the value of C and also calculate the mean of the distribution.
Answer & explanation
Correct answer: option 1
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 = $Σp_i{x_i}^2=C×0+2C×1+2C×2 + 3C×3+C^2×4 + 2C^2 × 5+ (7C^2 + C) × 6$
$=56C^2 + 21C = 56 ×\frac{1}{100}+ 21 ×\frac{1}{10}=\frac{266}{100}=\frac{133}{50}$