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Question
Choose the correct option from the given alternative:
If a d.r.v. X takes values 0, 1, 2, 3, . . . which probability P (X = x) = k (x + 1)·5 −x , where k is a constant, then P (X = 0) =
Options
`7/25`
`16/25`
`18/25`
`19/25`
Solution
If a d.r.v. X takes values 0, 1, 2, 3, . . . which probability P (X = x) = k (x + 1)·5 −x , where k is a constant, then P (X = 0) = `16/25`
Hint : `k[1/5^0 + 2/5^1 + 3/5^2 + ....]=1`
`Let s = k/5^0+ 2k/5^1+3k/5^2+ ...............`
i.e `s = k + 2k/5+3k/5^2+..........`
∴`1/5s = k/5 + (2k)/5^2 + (3k)/5^3 + ........`
∴ `s - 1/5s = k + k/5 + k/5^2 + k/5^3 + ..........`
∴ `4/5s=k[1 + 1/5 + 1/5^2 + 1/5^3 + ....]`
= `k [1/(1 - 1/5)] = 5k/4`
∴ `s = (25k)/16 = 1`
∴ `k = 16/25`
∴`P(x=0) = k(0+1)5^0 = k = 16/25.`
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