Mean and Standard Deviation of Binomial Distribution

A binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, each with the same probability of success. The mean and standard deviation of a binomial distribution can be calculated using the following formulas:

Mean = n * p
Standard Deviation = sqrt(n * p * (1-p))

where n is the number of trials and p is the probability of success in each trial.

Mode of Binomial Distribution

The mode of a probability distribution is the value that occurs most frequently. In a binomial distribution, the mode can be calculated as follows:

Mode = floor((n+1) * p)

where floor() is the floor function, which rounds down to the nearest integer.

Answer

Given that the mean and standard deviation of the binomial distribution are 5 and 2 respectively, we can solve for the values of n and p as follows:

Mean = n * p
5 = n * p

Standard Deviation = sqrt(n * p * (1-p))
2 = sqrt(n * p * (1-p))
4 = n * p * (1-p)

Using the first equation, we can solve for p as:

p = 5 / n

Substituting this into the third equation, we get:

4 = n * (5 / n) * (1 - (5/n))
4 = 5 - 25/n + 5/n
25/n = 1
n = 25

Substituting this back into the equation for p, we get:

p = 5/25 = 0.2

Therefore, the binomial distribution has n = 25 trials with a probability of success of p = 0.2.

Using the formula for the mode of a binomial distribution, we get:

Mode = floor((n+1) * p)
Mode = floor((25+1) * 0.2)
Mode = floor(5.2)
Mode = 5

Therefore, the mode of the binomial distribution is 5.