State whether the following is True or False : The p.m.f. of a r.v. X is P(x) = 2xn(n+1) , x = 1, 2, ……. n = 0 ,otherwise Then E(x) = 2n+13 - Mathematics and Statistics

MCQ

True or False

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)`

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.

Concept: Variance of Binomial Distribution (P.M.F.)

Is there an error in this question or solution?

Chapter 8 Probability Distributions
Miscellaneous Exercise 8 | Q 3.05 | Page 154