text: # 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)`
  
  Is there an error in this question or solution?
  Chapter 7: Probability Distributions - Miscellaneous Exercise 2 [Page 244]
  
  #### APPEARS IN
  
  Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
  Chapter 7 Probability Distributions
  Miscellaneous Exercise 2 | Q 10.3 | Page 244
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