Random Variables and Expected Value

Explore discrete distribution, expected value, net gain.

Lesson 3 of 53 parts Practice Flashcards

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Discrete Distribution

Objective: Explain discrete distribution and use it to solve or interpret a problem.

A discrete random variable assigns numbers to outcomes and a valid probability distribution has nonnegative probabilities summing to one. The distribution is invalid as stated.

Worked examples

Try itRandom Variables and Expected Value — practice 1.1: A game pays $15 with probability 1/5 and $0 otherwise. What is expected payout?Show answer

Step 1: Compute weighted average: (15)(1/5)+0(1−1/5). Step 2: The nonzero contribution is 15/5. Step 3: Expected payout is $3.

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Discrete Distribution

Objective: Explain discrete distribution and use it to solve or interpret a problem.

A discrete random variable assigns numbers to outcomes and a valid probability distribution has nonnegative probabilities summing to one. The distribution is invalid as stated.

Worked examples

Try itRandom Variables and Expected Value — practice 1.1: A game pays $15 with probability 1/5 and $0 otherwise. What is expected payout?Show answer

Step 1: Compute weighted average: (15)(1/5)+0(1−1/5). Step 2: The nonzero contribution is 15/5. Step 3: Expected payout is $3.

Expected Value

Objective: Explain expected value and use it to solve or interpret a problem.

E[X]=Σ xP(X=x) is a weighted average over outcomes, not a promise about the next trial. Include units such as dollars. No, one trial may have a different outcome.

Worked examples

Try itRandom Variables and Expected Value — practice 2.1: A game pays $32 with probability 1/8 and $0 otherwise. What is expected payout?Show answer

Step 1: Compute weighted average: (32)(1/8)+0(1−1/8). Step 2: The nonzero contribution is 32/8. Step 3: Expected payout is $4.

Net Gain

Objective: Explain net gain and use it to solve or interpret a problem.

Net expected gain equals expected payout minus the fixed cost of entry. A positive payout average may still produce negative net value after a fee. Gross payout is not the same as net gain.

Worked examples

Try itRandom Variables and Expected Value — practice 3.1: A game pays $22 with probability 1/11 and $0 otherwise, and costs $4 to play. Find expected net gain.Show answer

Step 1: Expected payout is (22)(1/11)=$2. Step 2: Subtract the fixed entry cost $4. Step 3: Expected net gain is $-2.

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