Question

If two independent random variable X and Y are binomially distributed with \(n_1=4\), p=(1/3) and \(n_2=5\), p= (1/3) respectively, Find the mode of the distribution of (X+Y).
1. 3, 2
2. 3
3. 3, 4
4. 4

• A. 3, 2
• B. 3
• C. 3, 4
• D. 4

Hint:

For any binomial distribution, the mode is always very close to the mean (\( \mu = np \)).
Here, the mean is \( 9 \times (1/3) = 3 \). This immediately suggests that the unique mode is \( 3 \).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem uses the reproductive (additive) property of independent binomial random variables to find the mode of their sum.

Step 2: Detailed Explanation:
Let us solve the problem systematically:
-

Step 1: Determine the distribution of the sum \( (X + Y) \):
We are given two independent binomial random variables:
\[ X \sim \text{Binomial}\left(n_1 = 4, p = \frac{1}{3}\right) \] \[ Y \sim \text{Binomial}\left(n_2 = 5, p = \frac{1}{3}\right) \] Since the probability of success \( p \) is identical for both independent variables, we can use the additive property of the binomial distribution.
Let \( W = X + Y \). The distribution of \( W \) is:
\[ W \sim \text{Binomial}(n, p) \] where:
\[ n = n_1 + n_2 = 4 + 5 = 9 \] \[ p = \frac{1}{3} \] So, \( W \sim \text{Binomial}\left(9, \frac{1}{3}\right) \).
- Determine the mode of the binomial distribution:
The mode of a binomial distribution \( B(n, p) \) is determined by calculating the value:
\[ (n + 1)p \] Substitute the parameters \( n = 9 \) and \( p = 1/3 \):
\[ (n + 1)p = (9 + 1) \times \frac{1}{3} = \frac{10}{3} = 3.333\dots \] - If \( (n+1)p \) is a non-integer, the distribution is unimodal, and the unique mode is the greatest integer less than or equal to \( (n+1)p \):
\[ \text{Mode} = \lfloor (n+1)p \rfloor = \lfloor 3.333\dots \rfloor = 3 \] - (If \( (n+1)p \) were an integer, say \( k \), the distribution would be bimodal with modes at \( k \) and \( k-1 \)).
Since \( 3.33 \) is not an integer, there is a single mode at \( 3 \).

Step 3: Final Answer:
The mode of the distribution of \( (X+Y) \) is \( 3 \).
Therefore, the correct choice is Option (B).