# Find the expected value, variance and standard deviation of the random variable whose p.m.f.’s are given below : x = x 1 2 3 ... n P (X = x) 1n 1n 1n ... 1n - Mathematics and Statistics

Sum

Find the expected value, variance and standard deviation of the random variable whose p.m.f.’s are given below :

 x = x 1 2 3 ... n P (X = x) 1/n 1/n 1/n ... 1/n

#### Solution

We construct the following table to find the expected value, variance, and standard deviation:

 xi P (xi) xi·P (xi) xi 2·P (xi) = xi × xi·P (xi) 1 1/n 1/n 1/n 2 1/n 2/n 2^2/n 3 1/n 3/n 3^2/n ... ... ... ... n 1/n n/n n^2/n

From the table,

∑xi . P(xi) = 1/n + 2/n + 3/n + ... + n/n

= 1/n ( 1 + 2 + 3 + ... + n)

= 1/n sum _( r=1)^n r = 1/n xx (n(n+1 ))/2

= (n+1)/2

∑xi2 . P(xi) = 1/n + 2^2/n + 3^2/n + ... + n^2/n

= 1/n ( 1^2 + 2^2 + 3^2 + ... + n^2)

= 1/n sum _( r=1)^n r^2 = 1/n xx (n(n+1 )(2n +1))/6

= ((n+1)(2n +1))/6

∴ expected value = E(X) = ∑ xi · P(xi)

= (n+1)/2

Variance = V(X) = ∑ xi2 . P (xi) - [ ∑ xi·P (xi) ]2

= ((n+1)(2n + 1))/6  - ((n + 1)/2)^2

= (n+1)/2 [(2n + 1)/ 3 -(n + 1) /2 ]

= (n+1)/2 [(4n+2-3n-3)/6 ]

=((n + 1)(n - 1))/12

= (n^2 - 1)/12

Standard deviation = sqrt (V(X)

= sqrt ((n^2-1)/12)

= (sqrt (n^2-1))/(2sqrt 3)`

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#### APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 7 Probability Distributions
Miscellaneous Exercise | Q 10.3 | Page 244