Discrete Random Variable Expected Value Exercises PDF

📚 Topic Summary

The expected value of a discrete random variable is the weighted average of all possible values of the variable. The weights are the probabilities of observing those values. It represents the average value you would expect to obtain if you repeated the experiment many times. Knowing how to calculate expected value is super useful in making decisions involving uncertainty!

For a discrete random variable $X$ with possible values $x_1, x_2, ..., x_n$ and corresponding probabilities $P(X=x_1), P(X=x_2), ..., P(X=x_n)$, the expected value, denoted as $E(X)$, is calculated as:

$E(X) = x_1P(X=x_1) + x_2P(X=x_2) + ... + x_nP(X=x_n)$

🔤 Part A: Vocabulary

Match the terms with their definitions:

1. Term: Probability Mass Function (PMF)
2. Term: Discrete Random Variable
3. Term: Expected Value
4. Term: Sample Space
5. Term: Outcome

1. Definition: The set of all possible results of a random experiment.
2. Definition: A variable whose value is a numerical outcome of a random phenomenon that can only take a finite number of values or a countably infinite number of values.
3. Definition: A function that gives the probability that a discrete random variable is exactly equal to some value.
4. Definition: The average value you would expect to obtain if you repeated the experiment many times.
5. Definition: A possible result of a random experiment.

✏️ Part B: Fill in the Blanks

Complete the following paragraph using the words provided:

(probabilities, random, expected value, discrete, average)

The _______ of a _______ variable is calculated by multiplying each possible value by its corresponding _______ and then summing the results. This gives the _______ or mean outcome we would expect over many trials. This is especially useful when working with _______ variables.

🤔 Part C: Critical Thinking

Explain, in your own words, how the expected value can be used to make informed decisions in real-life scenarios. Provide an example.