f(x,y)=8xy for 0
What will be the MGF of Y?
Do I need to insert the marginal density function for Y in the formula for MGF?
Also, what will be the limits? Will it be from x to 1 or from 0 to 1?
2025-01-13 08:02:55.1736755375
Moment Generating Function of Y
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From the question "Also, what will be the limits? Will it be from x to 1 or from 0 to 1?" I will assume that the proper joint density is $$ f_{X,Y}(x,y) = 8xy\cdot\mathsf 1_{0<x<y<1}. $$
To find the moment-generating function of $Y$, we first need to determine its marginal distribution. To do this, we integrate over all possible values of $X$:
$$ f_Y(y) = \int_0^y8xy\ \mathsf dx = 4y^3\cdot\mathsf 1_{(0,1)}(y). $$
Next we compute the expectation $$ \mathbb E[e^{tY}] = \int_0^1 e^{ty} 4y^3\ \mathsf dy = \frac{4 e^t (t ((t-3) t+6)-6)+24}{t^4} $$ (this requires some repeated integration by parts, which I have omitted for brevity.