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A card is drawn at random from a deck containing cards 2 ... 10.

$$ \Omega = \{2, 3, 4, ..., 10\} $$

A player draws a card from the deck and wins \$1 if the card is odd and loses \$1 if the card is even.

$$ X = \left\{ \begin{matrix} 1 & \mathrm{if\ odd} \\ -1 & \mathrm{if\ even} \end{matrix} \right. $$

What is the expected value of the payment?

How do I compute this using the definition of the expected value?

$$ E(X) = \sum_{x \in \Omega} x m(x) $$

what would my mass function be?

If I choose my mass function to be

$$ m_X(x) = \left\{ \begin{matrix} 5/9 & \mathrm{x\ is\ even} \\ 4/9 & \mathrm{x\ is\ odd} \end{matrix} \right. $$

Then by formula becomes

$$ E(X) = \left( (2+4+ ... + 10) \frac{5}{9}\right) + \left( (3+5+7+9)\frac{4}{9} \right) $$

which does not give me the right answer. I need the first term to be negative.

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Your representation of the expected value is not correct. Notice that your random variable is a function of the $x \in \Omega$ chosen: $$ X(x) = 1\ \text{if $x$ is odd} \ \text{and} \ X(x) = -1\ \text{if $x$ is even}.$$ Thus, you should write $$ E[X] =\sum_{x \in \Omega} X(x) m(x). $$ This is the expected value of $X$ with respect to the probability distribution that assigns mass $m(x)$ to point $x$.

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