Question

Given below are two statements.
Assertion (A):
For a random variable \(X\), if \(x=0,1,2,\ldots\) and \[ P(X=x)=k\frac{\lambda^x}{x!}, \] then for it to be a probability mass function, \(k=e^{-\lambda}\).
Reason (R):
In a probability distribution, the sum of all probabilities must be \(1\).

• A. Both (A) and (R) are correct and (R) is the correct explanation of (A)
• B. Both (A) and (R) are correct but (R) is not the correct explanation of (A)
• C. (A) is correct but (R) is not correct
• D. (A) is not correct but (R) is correct

Hint:

For a probability mass function, always use the condition \(\sum P(X=x)=1\).

Answer 1

The Correct Option is A

Concept:
For any probability mass function: \[ \sum P(X=x)=1 \]

Step 1: Use the given probability function. \[ P(X=x)=k\frac{\lambda^x}{x!} \]

Step 2: Sum over all possible values. \[ \sum_{x=0}^{\infty}P(X=x)=\sum_{x=0}^{\infty}k\frac{\lambda^x}{x!} \] \[ k\sum_{x=0}^{\infty}\frac{\lambda^x}{x!}=1 \]

Step 3: Use exponential series. \[ e^\lambda=\sum_{x=0}^{\infty}\frac{\lambda^x}{x!} \] So, \[ ke^\lambda=1 \] \[ k=e^{-\lambda} \] Therefore, Assertion is correct. Reason gives the rule used to derive \(k\), so it correctly explains Assertion. \[ \therefore \text{Correct Answer is (A)} \]