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Question
The mean and standard deviation of a binomial variate X are respectively 6 and 2. Find the probability mass function
Solution
X ~ B(n, p)
Given, Mean = 6
Standard deviation = 2
np = 6, `sqrt("npq")` = 2
npq = 4
`"npq"/"np"= 4/6`
q = `2/3`
p = 1 – q = `1/3`
npq = 4
`n" xx 1/3 xx 2/3` = 4
n = 18
Probability mass function
P(X = x) = ⁿCx px qn-x, x = 0, 1, 2, …. n
P(X = x) = `""^18"C"_x (1/3)^x (2/3)^(18 - x)`
x = 0, 1, 2, ..., 18
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