Answer:
![\mathbf{\dfrac{e^{2t}}{36} + \dfrac{e^{3t}}{18} + \dfrac{e^{4t}}{12} +\dfrac{e^{5t}}{9} + \dfrac{5e^{6t}}{36} + \dfrac{7e^{7t}}{6} + \dfrac{5e^{8t}}{36} + \dfrac{e^{9t}}{9} + \dfrac{e^{10t}}{12} + \dfrac{e^{11t}}{18} + \dfrac{e^{12t}}{36} }](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Cdfrac%7Be%5E%7B2t%7D%7D%7B36%7D%20%2B%20%5Cdfrac%7Be%5E%7B3t%7D%7D%7B18%7D%20%2B%20%5Cdfrac%7Be%5E%7B4t%7D%7D%7B12%7D%20%2B%5Cdfrac%7Be%5E%7B5t%7D%7D%7B9%7D%20%2B%20%20%5Cdfrac%7B5e%5E%7B6t%7D%7D%7B36%7D%20%2B%20%5Cdfrac%7B7e%5E%7B7t%7D%7D%7B6%7D%20%2B%20%5Cdfrac%7B5e%5E%7B8t%7D%7D%7B36%7D%20%2B%20%5Cdfrac%7Be%5E%7B9t%7D%7D%7B9%7D%20%2B%20%5Cdfrac%7Be%5E%7B10t%7D%7D%7B12%7D%20%2B%20%5Cdfrac%7Be%5E%7B11t%7D%7D%7B18%7D%20%2B%20%5Cdfrac%7Be%5E%7B12t%7D%7D%7B36%7D%20%7D)
Step-by-step explanation:
The objective is to find the moment generating function of
.
We are being informed that the fair die is rolled twice;
So; X to be the value for the first roll
Y to be the value of the second roll
The outcomes of X are: X = {1,2,3,4,5,6}
Where ;
![P (X=x) = \dfrac{1}{6}](https://tex.z-dn.net/?f=P%20%28X%3Dx%29%20%3D%20%20%5Cdfrac%7B1%7D%7B6%7D)
The outcomes of Y are: y = {1,2,3,4,5,6}
Where ;
![P (Y=y) = \dfrac{1}{6}](https://tex.z-dn.net/?f=P%20%28Y%3Dy%29%20%3D%20%20%5Cdfrac%7B1%7D%7B6%7D)
The outcome of Z = X+Y
![= \left[\begin{array}{cccccc}(1,1)&(1,2)&(1,3)&(1,4)&(1,5)&(1,6)\\ (2,1)&(2,2)&(2,3)&(2,4)&(2,5)&(2,6)\\ (3,1)&(3,2)&(3,3)&(3,4)&(3,5)&(3,6) \\ (4,1)&(4,2)&(4,3)&(4,4)&(4,5)&(4,6) \\ (5,1)&(5,2)&(5,3)&(5,4)&(5,5)&(5,6) \\ (6,1)&(6,2)&(6,3)&(6,4)&(6,5)&(6,6) \end{array}\right]](https://tex.z-dn.net/?f=%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccccc%7D%281%2C1%29%26%281%2C2%29%26%281%2C3%29%26%281%2C4%29%26%281%2C5%29%26%281%2C6%29%5C%5C%20%282%2C1%29%26%282%2C2%29%26%282%2C3%29%26%282%2C4%29%26%282%2C5%29%26%282%2C6%29%5C%5C%20%283%2C1%29%26%283%2C2%29%26%283%2C3%29%26%283%2C4%29%26%283%2C5%29%26%283%2C6%29%20%5C%5C%20%284%2C1%29%26%284%2C2%29%26%284%2C3%29%26%284%2C4%29%26%284%2C5%29%26%284%2C6%29%20%5C%5C%20%285%2C1%29%26%285%2C2%29%26%285%2C3%29%26%285%2C4%29%26%285%2C5%29%26%285%2C6%29%20%5C%5C%20%286%2C1%29%26%286%2C2%29%26%286%2C3%29%26%286%2C4%29%26%286%2C5%29%26%286%2C6%29%20%5Cend%7Barray%7D%5Cright%5D)
= [2,3,4,5,6,7,8,9,10,11,12]
Here;
![P (Z=z) = \dfrac{1}{36}](https://tex.z-dn.net/?f=P%20%28Z%3Dz%29%20%3D%20%20%5Cdfrac%7B1%7D%7B36%7D)
∴ the moment generating function
is as follows:
= ![E(e^{t(X+Y)}) = E(e^{tz})](https://tex.z-dn.net/?f=E%28e%5E%7Bt%28X%2BY%29%7D%29%20%3D%20E%28e%5E%7Btz%7D%29)
⇒ ![\sum \limits^{12}_ {z=2 } et ^z \ P(Z=z)](https://tex.z-dn.net/?f=%5Csum%20%5Climits%5E%7B12%7D_%20%7Bz%3D2%20%7D%20%20et%20%5Ez%20%5C%20P%28Z%3Dz%29)
= ![\mathbf{\dfrac{e^{2t}}{36} + \dfrac{e^{3t}}{18} + \dfrac{e^{4t}}{12} +\dfrac{e^{5t}}{9} + \dfrac{5e^{6t}}{36} + \dfrac{7e^{7t}}{6} + \dfrac{5e^{8t}}{36} + \dfrac{e^{9t}}{9} + \dfrac{e^{10t}}{12} + \dfrac{e^{11t}}{18} + \dfrac{e^{12t}}{36} }](https://tex.z-dn.net/?f=%5Cmathbf%7B%5Cdfrac%7Be%5E%7B2t%7D%7D%7B36%7D%20%2B%20%5Cdfrac%7Be%5E%7B3t%7D%7D%7B18%7D%20%2B%20%5Cdfrac%7Be%5E%7B4t%7D%7D%7B12%7D%20%2B%5Cdfrac%7Be%5E%7B5t%7D%7D%7B9%7D%20%2B%20%20%5Cdfrac%7B5e%5E%7B6t%7D%7D%7B36%7D%20%2B%20%5Cdfrac%7B7e%5E%7B7t%7D%7D%7B6%7D%20%2B%20%5Cdfrac%7B5e%5E%7B8t%7D%7D%7B36%7D%20%2B%20%5Cdfrac%7Be%5E%7B9t%7D%7D%7B9%7D%20%2B%20%5Cdfrac%7Be%5E%7B10t%7D%7D%7B12%7D%20%2B%20%5Cdfrac%7Be%5E%7B11t%7D%7D%7B18%7D%20%2B%20%5Cdfrac%7Be%5E%7B12t%7D%7D%7B36%7D%20%7D)