Suppose X is a binomial random variable with n 5 25 and p 5 0.80. a. Find the mean, variance, and standard deviation of X. b. Find the probability X is within one standard deviation of the mean. c. Find the probability X is more than two standard deviations from the mean

(a) The mean, variance and standard deviation of random variable X are 20, 4 and 2 respectively.

(b) The probability that X is within one standard deviation of the mean is 0.7926.

(c) The probability that X is more than two standard deviations from the mean is 0.0038.

Step-by-step explanation:

The random variable X follows a Binomial distribution with parameter n = 25 and p = 0.80.

The probability mass function of X is:

The mean, variance and standard deviation of a Binomial distribution is given by:

(a)

Compute the mean, variance and standard deviation of random variable X as follows:

Thus, the mean, variance and standard deviation of random variable X are 20, 4 and 2 respectively.

(b)

Compute the probability that X is within one standard deviation of the mean as follows:

P (μ - σ ≤ X ≤ μ + σ) = P (20 - 2 ≤ X ≤ 20 + 2)

                               = P (18 ≤ X ≤ 22)

                               = P (X ≤ 22) - P (X ≤ 18)

                              = P (X = 18) + P (X = 19) + P (X = 20) + P (X = 21) + P (X = 22)

Thus, the probability that X is within one standard deviation of the mean is 0.7926.

(c)

Compute the probability that X is more than two standard deviations from the mean as follows:

P (X > μ + 2σ) = P (X > 20 + (2×2))

                      = P (X > 24)

                      = P (X = 25)

Thus, the probability that X is more than two standard deviations from the mean is 0.0038.

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