Question

If $X$ is a binomial variate with the range 0, 1, 2, 3, 4, 5, 6 and $P(X=2)=4P(X=4),$ then the parameter $p$ of $X$ is

• A. $\frac{1}{3}$
• B. $\frac{1}{2}$
• C. $\frac{2}{3}$
• D. $\frac{3}{4}$

Hint:

In Binomial distribution, use the property $p + q = 1$ to solve for parameters once the ratio between $p$ and $q$ is found.

Answer 1

The Correct Option is A

Step 1: Identify Parameters
The range 0 to 6 indicates $n = 6$ for the binomial distribution. The probability mass function is $P(X=r) = {}^{n}C_{r} p^r q^{n-r}$.
Step 2: Set up Equation
Given $P(X=2) = 4P(X=4)$:
${}^{6}C_{2} p^2 q^4 = 4 \cdot {}^{6}C_{4} p^4 q^2$.
Step 3: Simplify
Since ${}^{6}C_{2} = {}^{6}C_{4} = 15$, the equation reduces to $q^2 = 4p^2$.
Taking the square root: $q = 2p$ (as $p, q>0$).
Step 4: Solve for $p$
Substitute $q = 1 - p$:
$1 - p = 2p \Rightarrow 3p = 1 \Rightarrow p = \frac{1}{3}$.
Final Answer: (a)