If moment generating function of continuous random variable X is \(\frac{λ}{λ-t}\) t < λ, then E(X3) equals to:

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SSC CGL Tier-II ( JSO ) 2021 Official Paper ( Held On: 10 August 2022 )
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  1. \(\frac{8}{\lambda^3}\)
  2. \(\frac{6}{\lambda^3}\)
  3. \(\frac{4}{\lambda^3}\)
  4. \(\frac{2}{\lambda^3}\)

Answer (Detailed Solution Below)

Option 2 : \(\frac{6}{\lambda^3}\)
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PYST 1: SSC CGL - English (Held On : 11 April 2022 Shift 1)
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Detailed Solution

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The correct answer is \(6/λ^3\)

Key Points

  • The moment generating function (MGF) of a random variable is a function that represents the expected values of its powers. 
    • The MGF of X is defined as M(t) = E(e^(tx)), where E represents the expected value.
    • The nth derivative of the MGF evaluated at t = 0 provides the expected value of X^n.

Additional Information

  • The condition -t < λ/λ ensures that the MGF of X exists and is finite.
  • The MGF is a useful tool to characterize the distribution of a random variable, and it provides a way to calculate its moments.

 Important Points

  • To calculate E(X^3) from the MGF of X, you need to take the third derivative of M(t) with respect to t, and then evaluate it at t = 0.
  • The MGF method assumes that the random variable X has a moment generating function, which may not be the case for all distributions.

 

 

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