Math Help - Moment Generating function of a normal variable

1. Moment Generating function of a normal variable

Hi everybody,

Again a problem with deriving an MGF.

I don't understand the manipulation in the first row. Why is e^xt disappearing? Why do we choose to make this (yellow marked) specific substitution....µ = µ + σ^2*t.

Thanks for your help!
[IMG]file:///C:/Users/Marmy/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]
[IMG]file:///C:/Users/Marmy/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]
[IMG]file:///C:/Users/Marmy/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]

2. Re: Moment Generating function of a normal variable

Originally Posted by Marmy
Hi everybody,

Again a problem with deriving an MGF.

I don't understand the manipulation in the first row. Why is e^xt disappearing? Why do we choose to make this (yellow marked) specific substitution....µ = µ + σ^2*t.

Thanks for your help! [IMG]file:///C:/Users/Marmy/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]
[IMG]file:///C:/Users/Marmy/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]
[IMG]file:///C:/Users/Marmy/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif[/IMG]
The e^(xt) has not disappeared. Expand the exponent in the second integral and it clearly appears.

The reason for the substitution is clear from the entire solution ....

3. Re: Moment Generating function of a normal variable

Thanks,

I managed to understand this problem now!