Maths-
General
Easy

Question

If X is a random variable with the following distribution  where p+q=1 then the variance of X is

  1. ap+bq    
  2. ap‐bq    
  3. p q left parenthesis a minus b right parenthesis to the power of 2 end exponent    
  4. p q left parenthesis a plus b right parenthesis to the power of 2 end exponent    

The correct answer is: p q left parenthesis a minus b right parenthesis to the power of 2 end exponent


    Mean o f X equals mu equals not stretchy sum subscript blank superscript blank x subscript i end subscript P left parenthesis X equals x subscript i end subscript right parenthesis equals a p plus b q variance of X equals 0 to the power of 2 end exponent equals not stretchy sum subscript blank superscript blank x subscript i end subscript superscript 2 end superscript P left parenthesis X equals x subscript j end subscript right parenthesis minus mu to the power of 2 end exponent
    equals a to the power of 2 end exponent p plus b to the power of 2 end exponent q minus left parenthesis a p plus b q right parenthesis to the power of 2 end exponent
    equals a to the power of 2 end exponent p plus b to the power of 2 end exponent q minus a to the power of 2 end exponent p to the power of 2 end exponent minus b to the power of 2 end exponent q to the power of 2 end exponent minus 2 a b p q
    equals a to the power of 2 end exponent p left parenthesis 1 minus p right parenthesis plus b to the power of 2 end exponent q left parenthesis 1 minus q right parenthesis minus 2 a b p q
    equals a to the power of 2 end exponent p q plus b to the power of 2 end exponent q p minus 2 a b p q equals p q left parenthesis a to the power of 2 end exponent plus b to the power of 2 end exponent minus 2 a b right parenthesis
    equals p q left parenthesis a minus b right parenthesis to the power of 2 end exponent

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