# How would a moment generating function change if all the random variables are increased by a value

Suppose you have some moment generating function $$M_x(t)$$ Now all the random variables x are increased by a arbitrary value b. What is the new moment generating value?

I tried solving this by moving back from the MGF to the probability distribution, but this proved impossible. Could somebody give me a nudge in the right direction? I'm looking for a 'proven' method instead of using intuition. Thanks!

Edit:

Is this correct?

Declare a new variable $$y = x + b$$, so $$x = y - b$$

$$E(e^{tx}) = E(e^{ty-tb}) = E(e^{ty})E(e^{-tb}) = M_y(t)E(e^{-tb}) =M_y(t)e^{-tb}$$

• Can you clarify? You use $x$ as argument to the mgf $M(x)$ and then name it all the random variables x, but the argument to the mgf is not the random variable! Sep 25 '20 at 13:56
• Yes I get that. Let me rephrase it: I have some random variable x and the moment generating function of x is M(t) Sep 25 '20 at 13:59
Declare a new variable $$y = x + b$$, so $$x = y - b$$
$$E(e^{tx}) = E(e^{ty-tb}) = E(e^{ty})E(e^{-tb}) = M_y(t)E(e^{-tb}) =M_y(t)e^{-tb}$$