Discrete Distributions

Learn discrete distributions, events for a random variable.

Lesson 1 of 32 parts Practice Flashcards

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Discrete Distributions

Objective: Explain discrete distributions, apply the stated method, and check its conditions.

A discrete random variable assigns numerical values to outcomes, with nonnegative probabilities summing to one. Outcomes can share a variable value. A valid probability table must cover all possible values and assign total probability one before calculating summaries. Both nonnegativity and a total of one are required.

Formulas

$ΣP(X=x)=1; P(X=x)≥0$

Worked examples

Try itA discrete variable has probabilities 6/20, 7/20, and p for its three outcomes. Find p.Show answer

Step 1: Probabilities in a valid distribution sum to one. Step 2: p=1−6/20−7/20. Step 3: p=7/20, which is nonnegative.

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Discrete Distributions

Objective: Explain discrete distributions, apply the stated method, and check its conditions.

Formulas

$ΣP(X=x)=1; P(X=x)≥0$

Worked examples

Try itA discrete variable has probabilities 6/20, 7/20, and p for its three outcomes. Find p.Show answer

Step 1: Probabilities in a valid distribution sum to one. Step 2: p=1−6/20−7/20. Step 3: p=7/20, which is nonnegative.

Events for a Random Variable

Objective: Explain events for a random variable, apply the stated method, and check its conditions.

To calculate P(X∈S) for a discrete variable, add the probabilities of every listed value in S. Distinct variable values are disjoint outcomes. Translate inequalities carefully, including whether endpoints are included, before adding table entries. Check inclusive versus strict inequalities at event boundaries.

Formulas

$P(X∈S)=Σₓ∈S P(X=x)$

Worked examples

Try itA random variable takes values 0, 1, 2 with probabilities 1/7, 1/7, and 5/7. Find P(X≥1).Show answer

Step 1: The event includes outcomes 1 and 2. Step 2: Add 1/7+5/7. Step 3: The probability is 6/7.

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