If the mean and variance of a binomial distribution are 5 and 4, respectively, then the value of n is:

The correct answer is 25

Key PointsGiven:

Mean = np =5

Variance = np (1−p) = 4

Calculation:

Variance = 4

⇒ np (1−p) = 4

⇒ 1 − p = 4/5​

⇒ 1 − p = 4/5

⇒ p = 1/5 

Since, np = 5

⇒ n × 1/5 = 5

⇒ n = 25

∴ The value of n is 25.

Given the mean and variance of a binomial distribution, we can find the values of n and p. The mean of a binomial distribution is equal to n * p, and the variance is equal to n * p * (1 - p). We can use these two equations to solve for n and p.

Additional Information 

• The binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials.
• The parameters of a binomial distribution are n (the number of trials) and p (the probability of success in each trial).
• The mean and variance of a binomial distribution are n * p and n * p * (1 - p), respectively.
• Given the mean and variance of a binomial distribution, we can use these two equations to find the values of n and p.
• Understanding the properties of binomial distributions and how to find the mean and variance is important in many areas of statistics and data analysis.