23  Expected Values for Continuous Random Variables

Example 23.1 Continuing Example 22.1. Han’s arrival time \(X\) (minutes after noon) has pdf \[ f_X(x) = \begin{cases} (2/3600)x, & 0<x<60,\\ 0 & \text{otherwise.} \end{cases} \]

1. Explain how you could use simulation to approximate \(\text{E}(X)\).
   
   
   
   
   
2. Explain how you could approximate \(\text{E}(X)\) as a sum (as if \(X\) were a discrete random variable).
   
   
   
   
   
3. How could you obtain a better approximation than the sum in the previous part? What happens in the limit?
   
   
   
   
   
4. Compute and interpret \(\text{E}(X)\).
   
   
   
   
   
5. Compute and interpret \(\text{P}(X \le \text{E}(X))\).
   
   
   
   
   

• The expected value (a.k.a. expectation a.k.a. mean), of a random variable \(X\) is a number denoted \(\text{E}(X)\) representing the probability-weighted average value of \(X\).
• The expected value of a continuous random variable with pdf \(f_X\) is defined as \[ \text{E}(X) = \int_{-\infty}^\infty x f_X(x) dx \]
• Replace the generic bounds \((-\infty, \infty)\) with the possible values of the random variable
• Note well that \(\text{E}(X)\) represents a single number.
• The expected value is the “balance point” (center of gravity) of a distribution.
• The expected value of a random variable \(X\) is defined by the probability-weighted average according to the underlying probability measure. But the expected value can also be interpreted as the long-run average value, and so can be approximated via simulation.
• Read the symbol \(\text{E}(\cdot)\) as
  • Simulate lots of values of what’s inside \((\cdot)\)
  • Compute the average. This is a “usual” average, regardless of whether the variable is discrete or continuous; just sum all the simulated values and divide by the number of simulated values.

Example 23.2 Continuing Example 22.3. Let \(X\) be the times (minutes) until the next earthquake occurs (of any magnitude in a certain region). Assume that the pdf of \(X\) is \[ f_X(x) = (1/120) e^{-x/120}, \qquad x \ge0 \]

1. Explain how you could use simulation to approximate \(\text{E}(X)\).
   
   
   
   
   
2. Donny Dont says \(\text{E}(X) = \int_0^\infty (1/120)e^{-x/120}dx = 1\). Do you agree?
   
   
   
   
   
3. Compute and interpret \(\text{E}(X)\).
   
   
   
   
   
4. Compute and interpret \(\text{P}(X \le \text{E}(X))\).
   
   
   
   
   

Example 23.3 Continuing Example 23.1. Han’s arrival time \(X\) (minutes after noon) has pdf \[ f_X(x) = \begin{cases} (2/3600)x, & 0<x<60,\\ 0 & \text{otherwise.} \end{cases} \]

1. Explain how you could use simulation to approximate \(\text{E}(X^2)\).
   
   
   
   
   
2. Explain how you could approximate \(\text{E}(X^2)\) as a sum (like a discrete random variable).
   
   
   
   
   
3. How could you obtain a better approximation than the sum in the previous part? What happens in the limit?
   
   
   
   
   
4. Compute \(\text{E}(X^2)\).
   
   
   
   
   
5. Compute \(\text{Var}(X)\).
   
   
   
   
   
6. Compute \(\text{SD}(X)\).
   
   
   
    

• The “law of the unconscious statistician” (LOTUS) says that the expected value of a transformed random variable can be found without finding the distribution of the transformed random variable, simply by applying the probability weights of the original random variable to the transformed values. \[\begin{align*} & \text{Discrete $X$ with pmf $p_X$:} & \text{E}[g(X)] & = \sum_x g(x) p_X(x)\\ & \text{Continuous $X$ with pdf $f_X$:} & \text{E}[g(X)] & =\int_{-\infty}^\infty g(x) f_X(x) dx \end{align*}\]
• LOTUS says we don’t have to first find the distribution of \(Y=g(X)\) to find \(\text{E}[g(X)]\); rather, we just simply apply the transformation \(g\) to each possible value \(x\) of \(X\) and then apply the corresponding weight for \(x\) to \(g(x)\).
• Whether in the short run or the long run, in general \[\begin{align*} \text{Average of $g(X)$} & \neq g(\text{Average of $X$}) \end{align*}\]
• In terms of expected values, in general \[\begin{align*} \text{E}(g(X)) & \neq g(\text{E}(X)) \end{align*}\] The left side \(\text{E}(g(X))\) represents first transforming the \(X\) values and then averaging the transformed values. The right side \(g(\text{E}(X))\) represents first averaging the \(X\) values and then plugging the average (a single number) into the transformation formula.

Example 23.4 Continuing Example 23.2. Let \(X\) be the times (minutes) until the next earthquake occurs (of any magnitude in a certain region). Assume that the pdf of \(X\) is \[ f_X(x) = (1/120) e^{-x/120}, \qquad x \ge0 \]

1. Explain how you could use simulation to approximate \(\text{E}(X^2)\).
   
   
   
   
   
2. Donny Dont says: “I can just use LOTUS and replace \(x\) with \(x^2\), so \(\text{E}(X^2)\) is \(\int_{-\infty}^{\infty} (1/120)x^2 e^{-x^2/120} dx\)”. Do you agree?
   
   
   
   
   
3. Compute \(\text{E}(X^2)\).
   
   
   
   
   
4. Compute \(\text{Var}(X)\).
   
   
   
    
5. Compute \(\text{SD}(X)\).