Question 1197967
**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!