๐ Topic Summary
The expected value of a function of a random variable $X$, denoted as $E[g(X)]$, represents the average value of the function $g(X)$ over all possible values that $X$ can take. Instead of calculating the expected value of $X$ directly, we apply the function $g$ to each possible outcome of $X$ and then compute the weighted average, using the probability distribution of $X$ as the weights. This concept is crucial in various fields, including finance, insurance, and decision theory, allowing us to make informed predictions and decisions under uncertainty.
In simpler terms, if you have a random number generator $X$, and you want to know what the average output of some formula $g(X)$ would be if you ran the generator many times, that's what the expected value $E[g(X)]$ tells you. You don't need to actually run the generator many times; you just need to know the probabilities of each possible output of $X$.
๐งฎ Part A: Vocabulary
Match the terms with their correct definitions.
| Term |
Definition |
| 1. Random Variable |
A. A function that transforms a random variable. |
| 2. Expected Value |
B. The average value of a function over all possible outcomes. |
| 3. Probability Distribution |
C. A variable whose value is a numerical outcome of a random phenomenon. |
| 4. Function of a Random Variable |
D. A description of how probabilities are distributed over the values of a random variable. |
| 5. $E[g(X)]$ |
E. The notation for the expected value of a function of X. |
Match the following: 1-C, 2-B, 3-D, 4-A, 5-E
โ๏ธ Part B: Fill in the Blanks
Complete the following paragraph using the words: function, probabilities, weighted average, random variable, $E[g(X)]$.
The expected value of a __________ of a __________ , represented by __________, is calculated by taking the __________ of the function's values, using the __________ of the outcomes as weights.
Answer: The expected value of a function of a random variable, represented by $E[g(X)]$, is calculated by taking the weighted average of the function's values, using the probabilities of the outcomes as weights.
๐ค Part C: Critical Thinking
Explain, in your own words, why understanding the expected value of a function of a random variable is important in real-world decision-making. Provide a specific example where this concept could be applied.