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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)`

Options

  • True

  • False

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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.

Concept: Variance of Binomial Distribution (P.M.F.)
  Is there an error in this question or solution?
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 8 Probability Distributions
Miscellaneous Exercise 8 | Q 3.05 | Page 154
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