SOLUTION: Let {{{ M_X (t) }}} be the moment generating function for a random variable X. Which of the following statements about {{{ M_X (t) }}} are true? I {{{ M_X (0) }}} = 1 II {{{ ((d

Algebra ->  Probability-and-statistics -> SOLUTION: Let {{{ M_X (t) }}} be the moment generating function for a random variable X. Which of the following statements about {{{ M_X (t) }}} are true? I {{{ M_X (0) }}} = 1 II {{{ ((d      Log On


   

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