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Question
State whether the following is True or False :
The p.m.f. of a r.v. X is P(x) = `(2x)/("n"("n" + 1))` , x = 1, 2, ……. n
= 0 ,otherwise
Then E(x) = `(2"n" + 1)/(3)`
Options
True
False
Solution
| X | 1 | 2 | 3 | .... | n |
| P(X) | `(2)/("n"("n" + 1)` | `(4)/("n"("n" + 1)` | `(6)/("n"("n" + 1)` | .... | `(2"n")/("n"("n" + 1)` |
E(X) = `sumx_1*"p"(x_"i")`
= `1.(2)/("n"("n" + 1)) + 2.(4)/("n"("n" + 1)) + 3.(6)/("n"("n" + 1)) + .... + "n". (2"n")/("n"("n" + 1)`
= `(2"n")/("n"("n" + 1))(1 + 4 + 9 + ... + "n"^2)`
= `(2"n")/("n"("n" + 1))(1^2 + 2^2 + 3^2 + ... + "n"^2)`
= `(2"n")/("n"("n" + 1))("n"("n" + 1)(2"n" + 1))/(6)`
= `(2"n" + 1)/(3)` is True.
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