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प्रश्न
In a binomial distribution consisting of 5 independent trials, the probability of 1 and 2 successes are 0.4096 and 0.2048 respectively. Find the mean and variance of the random variable
उत्तर
n = 5, X ~ B(n, p)
Given, P(X = 1) = 0.4096
P(X = 2) = 0.2048
P(X = x) = ⁿCx px qn-x, x = 0, 1, 2, …. n
⇒ 5C1 pq4 = 0.4096
⇒ 5C2 p2q3 = 0.2048
5pq4 = 0.4096 .......(1)
10p2q3 = 0.2048 .......(2)
(1) divided by (2),
⇒ `(5"pq"^4)/(10"p"^2"q"^3)` = 2
`"p"/"q"` = 4
q = 4p
q = 4(1 – q)
q = 4 – 4q
5q = 4
q = `4/5`
p = 1 – q = `1/5`
Mean = np
= `5 xx 1/5` = 1
Variance = npq
= `1 xx 4/5 = 4/5`
Distribution
P(X = x) = `""^5"C"_x (1/5)^x (4/5)^(5-x)`
x = 0, 1, 2, 3, 4, 5
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