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Let X be the score on rolling a fair die. Calculate $E(a^X)$ where a is a real constant.

I don't even know where to start?

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    Check out Moment generating functions2012-11-25

2 Answers 2

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$Y=a^X$ is a random variable that is $a$ with probability $1/6$, $a^2$ with probability $1/6$, $a^3$ with probability $1/6$, $a^4$ with probability $1/6$, $a^5$ with probability $1/6$, and $a^6$ with probability $1/6$. What is $E(Y)$? Since you know $Y$'s possible values and probabilities exactly you can forget where they came from, and just calculate its expectation right away.

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    Thanks, I completely understand now.2012-11-25
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Hint The definition of the expected value for a function $g(X)$ of random variable is $ \sum_{k\in\Omega} p(k)g(k) $ in the discrete case, and $ \int_{\Omega} f(x)g(x)\text{d}x $ in the continuous case, where $p(k)$ is the probability mass function, and $f(x)$ the probability density function.

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    p(k) is the probability mass function, and g(k) in your case is a^k, the function for which you want to take the expected value2012-11-25