Question

Which of the following is NOT true?

• A. Poisson distribution with mean and variance as 16
• B. Binomial distribution with mean 16 and variance 6
• C. The contingency table follows multinomial distribution
• D. Binomial distribution with mean 16 and standard deviation 4

Hint:

Exam Tip:
For binomial distribution:

• Mean \(= np\).
• Variance \(= np(1-p)\).
• Variance \(<\) Mean (unless \(p = 0\) or \(p = 1\)).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to check the validity of each statement based on the properties of distributions.

Step 2: Analyzing Each Option:

• (A) Poisson distribution with mean and variance as 16:
For Poisson distribution, mean = variance.
If mean = 16, then variance = 16.
So, this is true.
• (B) Binomial distribution with mean 16 and variance 6:
For a binomial distribution, variance \(= np(1-p)\).
Mean \(= np = 16\).
Variance \(= np(1-p) = 16(1-p)\).
For variance to be 6, we need \(16(1-p) = 6 \Rightarrow 1-p = 6/16 = 0.375 \Rightarrow p = 0.625\).
Since \(0 < p < 1\), this is possible.
So, this could be true.
Let's check if \(p = 0.625\) gives variance 6.
\(16 \times 0.625 \times 0.375 = 16 \times 0.234375 = 3.75\), not 6.
Wait, variance \(= np(1-p) = 16(1-p)\).
For variance = 6, \(1-p = 6/16 = 0.375 \Rightarrow p = 0.625\).
Then variance = \(16 \times 0.625 \times 0.375 = 3.75\), not 6.
The formula is correct: variance = npq.
If mean = 16 and variance = 6, then \(p = 1 - \frac{\text{variance}}{\text{mean}} = 1 - \frac{6}{16} = \frac{10}{16} = 0.625\).
Then \(q = 0.375\).
Variance = \(16 \times 0.625 \times 0.375 = 3.75\), not 6.
So, it is not possible to have mean 16 and variance 6 for a binomial distribution.
This statement is false.
• (C) The contingency table follows multinomial distribution:
This is true. A contingency table with fixed row and column totals follows a multinomial distribution.
• (D) Binomial distribution with mean 16 and standard deviation 4:
Standard deviation = 4, so variance = 16.
For binomial, mean = variance = 16 implies \(p = 1/2\).
This is possible. So, this is true.

Step 3: Final Answer:
Therefore, option (B) is NOT true.