# Expected values and continuous distributions

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1. Recall (or learn!) that for a discrete random variable X taking values in a finite set S, the expected value of X is defined as

$\displaystyle \mathbb{E}[X] = \sum_{x \in S} x \mathbb{P}[X = x]$ .

Two fair, six-sided dice are thrown, and the scores $\displaystyle (X_1, X_2)$ are recorded. Find the expected value of each of the following quantities:

1. $\displaystyle Y = X_1 + X_2$ , the sum of the scores.
2. $\displaystyle M = \frac{1}{2}(X_1+X_2)$ , the average of the scores.
3. $\displaystyle Z = X_1 X_2$ , the product of the scores.
4. $\displaystyle U = \min\{X_1, X_2\}$ , the minimum score
5. $\displaystyle V = \max\{X_1, X_2\}$ , the maximum score.

2. For a continuous random variable X with probability density function f taking values in S, the expected value of X is defined as

$\displaystyle \mathbb{E}[X] = \int_S x f(x) dx$ .

Suppose that a random variable X has the probability density function

$\displaystyle f(x) = r e^{-rx}, 0 \leq x < \infty$ for some r > 0.

(This is called the exponential distribution with rate r.)

1. Confirm that f(x) is a valid probability density function.
2. Compute the expected value of X.

(exercises from the excellent http://www.math.uah.edu/stat/)