Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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?

share|improve this question
Check out Moment generating functions –  Gautam Shenoy Nov 25 '12 at 15:10
add comment

2 Answers

up vote 0 down vote accepted

$Y=a^X$ 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.

share|improve this answer
I think I see what you mean. Am I right in thinking E(Y)=3.5? –  Mathlete Nov 25 '12 at 15:30
No. $E(X)=3.5$, but $E(Y)=\frac16(a+a^2+a^3+a^4+a^5+a^6)$. –  Hagen von Eitzen Nov 25 '12 at 15:42
Thanks, I completely understand now. –  Mathlete Nov 25 '12 at 15:47
add comment

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.

share|improve this answer
What are p(k) and g(k)? –  Mathlete Nov 25 '12 at 15:12
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 value –  Jean-Sébastien Nov 25 '12 at 15:14
add comment

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.