Question: b) If \( x \) is a binomial random variable with \( n = 12 \) and \( p = 0.25 \) i. Find the…

b) If \( x \) is a binomial random variable with \( n = 12 \) and \( p = 0.25 \)

i. Find the mean value of \( x \). (Mean = np)

ii. Find the standard deviation of \( x \). (Standard deviation = \(\sqrt{npq}\))

Solution

b) If \( x \) is a binomial random variable with \( n = 12 \) and \( p = 0.25 \) i. Find the mean value of \( x \). The formula for the mean of a binomial distribution is \[ \mu = np \] Substituting the given values, we get \[ \mu = 12 \times 0.25 \] \[ \mu = 3 \] ii. Find the standard deviation of \( x \). The formula for the standard deviation of a binomial distribution is \[ \sigma = \sqrt{npq} \] where \( q = 1 - p \). First, calculate \( q \): \[ q = 1 - 0.25 = 0.75 \] Now calculate the standard deviation: \[ \sigma = \sqrt{12 \times 0.25 \times 0.75} \] \[ \sigma = \sqrt{2.25} \] \[ \sigma = 1.5 \]