If moment generating function is $m(t)=[(1/3)e^{t}+(2/3)]^{5}$,
then what is the distributions of the random variable?
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Maybe work backwards. The mgf of a sum of independent random variables is the product of the mgf. So our mgf is the mgf of a sum $X_1+X_2+\cdots +X_5$ of $5$ independent random variables with mgf $(1/3)e^t +2/3$. Now do we recognize this? Hint: What is the mgf of a random variable which is $1$ with probability $p$ and $0$ with probability $1-p$?
André Nicolas
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thanks~ Binomial distribution~ – LeeDH Apr 19 '14 at 15:42
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Prof. Nicolas would you see this, please? I really hope you can answer it for me. Thank you. (-^〇^-) – Anastasiya-Romanova 秀 Apr 19 '14 at 16:36
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Got back, saw it. The answer that was given is correct. For fixed $j$, the inner limit is $0$, so the full limit is $0$. – André Nicolas Apr 19 '14 at 19:54