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  # The Random Variable X Can Take Only the Values 0, 1, 2, 3. Give that P(X = 0) = P(X = 1) = P and P(X = 2) = P(X = 3) Such that Sigmap_I X_I^2 = 2sigmap_Ix_I. Find the Value of P - CBSE (Science) Class 12 - Mathematics
  
  #### Question
  
  The random variable X can take only the values 0, 1, 2, 3. Give that P(X = 0) = P(X = 1) = p and P(X = 2) = P(X = 3) such that Sigmap_i x_i^2 = 2Sigmap_ix_i. Find the value of p
  
  #### Solution
  
  It is given that the random variable X can take only the values 0, 1, 2, 3.
  
  Given:
  
  P(X = 0) = P(X = 1) = p
  
  P(X = 2) = P(X = 3)
  
  Let P(X = 2) = P(X = 3) = q
  
  Now,
  
  P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 1
  
  => p + p + (q + q) = 1
  
  => q = (1- 2p)/2
  
  Since, Sigmap_ix_i^2 = 2Sigmap_ix_i
  
  => 0 + p(1)^2 + ((1-2p)/2)(2^2 + 3^2) = 2[0 + p + ((1-2p)/2) (2 + 3)]
  
  => p + 13/2 (1- 2p) = 2[p + 5/2(1-2p)]
  
  => p + 13/2 - 13p = 2p + 5 - 10p
  
  => 13/2 - 12p = -8p + 5
  
  => 4p = 3/2
  
  => p = 3/8
  
  Is there an error in this question or solution?
  
  #### APPEARS IN
  
  Solution The Random Variable X Can Take Only the Values 0, 1, 2, 3. Give that P(X = 0) = P(X = 1) = P and P(X = 2) = P(X = 3) Such that Sigmap_I X_I^2 = 2sigmap_Ix_I. Find the Value of P Concept: Variance of a Random Variable.
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