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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{E}[X] = \sum_{x \in S} x \mathbb{P}[X = x]} .

Two fair, six-sided dice are thrown, and the scores Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (X_1, X_2)} are recorded. Find the expected value of each of the following quantities:

  1. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y = X_1 + X_2} , the sum of the scores.
  2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M = \frac{1}{2}(X_1+X_2)} , the average of the scores.
  3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z = X_1 X_2} , the product of the scores.
  4. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle U = \min\{X_1, X_2\}} , the minimum score
  5. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{E}[X] = \int_S x f(x) dx } .

Suppose that a random variable X has the probability density function

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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/)