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Question
Solve the following problem :
If X follows Poisson distribution with parameter m such that
`("P"("X" = x + 1))/("P"("X" = x)) = (2)/(x + 1)`
Find mean and variance of X.
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Solution
Given, X ~ P(m) and `("P"("X" = x + 1))/("P"("X" = x)) = (2)/(x + 1)`
The p.m.f. of X is given by
P(X = x) = `("e"^(-"m")"m"^x)/(x!)`
∴ According to the given condition, we get
`(("e"^(-"m")"m"^(x + 1))/((x + 1)!))/(("e"^(-"m")"m"^x)/(x!)) = (2)/(x + 1)`
∴ `("e"^(-"m") xx "m"^x xx "m")/((x + 1) xx x!) xx (x!)/("e"^(-"m") xx "m"^x) = (2)/(x + 1)`
∴ `"m"/(x + 1) = (2)/(x + 1)`
∴ m = 2
∴ Mean = Variance = m = 2.
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