Question

If the random variable $X$ takes the values $X_1,X_2,................X_{10}$ with probabilities $P_{x_i}=Ki$ then the value of $K$ is

• A. $\frac{1}{10}$
• B. $\frac{1}{4}$
• C. $\frac{1}{55}$
• D. $\frac{7}{12}$
• E. $\frac{3}{4}$

Answer 1

The correct option is C

$\frac{1}{55}$

Explanation of the correct option.

Given: Random variable $X$ takes the values $X_1,X_2,................X_{10}$.

Therefore, $X=X_i$, $P\left(X_i\right)=Ki$

$P\left(X_1\right)=K,P\left(X_2\right)=2K,...............P\left(X_{10}\right)=10K$

We know that the sum of probabilities is equal to $1$.

$K+2K+3K+4K+5K+6K+7K+8K+9K+10K=1$

$\Rightarrow$ $\left(1+2+3+4+5+6+7+8+9+10\right)K=1$

$\Rightarrow$ $\frac{10\left(10+1\right)}{2}K=1$ $[\because$The sum of the first $n$ natural number is given by $\frac{n\left(n+1\right)}{2}$$]$

$\Rightarrow$ $55K=1$

$\Rightarrow$ $K=\frac{1}{55}$

Hence option $\left(C\right)$ is the correct option.