document.write( "Question 1197967: Let \"+M_X+%28t%29+\" be the moment generating function for a random variable X. Which
\n" ); document.write( "of the following statements about \"+M_X+%28t%29+\" are true?
\n" ); document.write( "I \"+M_X+%280%29+\" = 1
\n" ); document.write( "II \"+%28%28d%5E2%29+M_X%28t%29%29%2F%28d+t%5E2%29+\" for t=0 = Var(X)
\n" ); document.write( "III M_X(t) uniquely determines the probability distribution for X.
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Algebra.Com's Answer #848349 by onyulee(41)\"\" \"About 
You can put this solution on YOUR website!
**I. M_X(0) = 1**\r
\n" ); document.write( "\n" ); document.write( "* **True.**
\n" ); document.write( " * By definition, the moment-generating function (MGF) of a random variable X is given by:
\n" ); document.write( " * M_X(t) = E[e^(tX)]
\n" ); document.write( " * When t = 0:
\n" ); document.write( " * M_X(0) = E[e^(0*X)] = E[e^0] = E[1] = 1
\n" ); document.write( " * Since the expected value of a constant (1) is 1, M_X(0) always equals 1.\r
\n" ); document.write( "\n" ); document.write( "**II. d^2 M_x(t)/dt^2 for t=0 = Var(X)**\r
\n" ); document.write( "\n" ); document.write( "* **True.**
\n" ); document.write( " * The second derivative of the MGF evaluated at t = 0 gives the variance of the random variable. \r
\n" ); document.write( "\n" ); document.write( "**III. M_X(t) uniquely determines the probability distribution for X.**\r
\n" ); document.write( "\n" ); document.write( "* **Generally True.**
\n" ); document.write( " * If two random variables have the same moment-generating function, then they have the same probability distribution.
\n" ); document.write( " * However, there are some rare exceptions where different distributions can have the same MGF in a small interval around t = 0.\r
\n" ); document.write( "\n" ); document.write( "**In summary:**\r
\n" ); document.write( "\n" ); document.write( "* **I and II are always true.**
\n" ); document.write( "* **III is generally true, with some minor exceptions.**\r
\n" ); document.write( "\n" ); document.write( "Let me know if you'd like to explore any of these points further!
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