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Nastasia [14]
3 years ago
5

Let X and Y be random variables with the following joint moment generating function: M_{X,Y}(s,t)=0.1+0.2e^s+0.3e^t+0.4e^{s+t}

Mathematics
1 answer:
MA_775_DIABLO [31]3 years ago
8 0

We obtain the joint PMF directly from the joint MGF:

M_{X,Y}(s,t)=0.1+0.2e^s+0.3e^t+0.4e^{s+t}

\implies\mathrm{Pr}[X=x,Y=y]=\begin{cases}0.1&\text{for }x=y=0\\0.2&\text{for }x=1,y=0\\0.3&\text{for }x=0,y=1\\0.4&\text{for }x=y=1\\0&\text{otherwise}\end{cases}

Then

\mathrm{Pr}[X=Y]=\mathrm{Pr}[X=Y=0]+\mathrm{Pr}[X=Y=1]=0.1+0.4=\boxed{0.5}

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