Probability Distributions

Grade 11 · Math · Study Guide

An exploration of discrete and continuous probability distributions, expected value, and key distribution models.

A probability distribution describes how the probabilities are distributed over the values of a random variable. A discrete random variable takes on countable values; its probability distribution is a probability mass function (PMF) where all probabilities sum to 1. A continuous random variable takes on values in an interval; its distribution is described by a probability density function (PDF), and probabilities are found as areas under the curve.

The expected value (mean) of a discrete random variable is E(X) = Σ x·P(X=x). The variance is Var(X) = E(X²) - [E(X)]² and measures spread. The standard deviation is the square root of variance.

Key distributions include: the Binomial Distribution, which models the number of successes in n independent trials each with probability p of success: P(X=k) = C(n,k)·p^k·(1-p)^(n-k). The mean is np and variance is np(1-p). The Normal Distribution is continuous, bell-shaped, and symmetric about its mean μ; about 68% of values fall within one standard deviation, 95% within two, and 99.7% within three (the empirical rule). The standard normal distribution has μ=0 and σ=1, and z-scores transform any normal variable: z = (x - μ)/σ.

Key terms

Random Variable

A variable whose value is determined by the outcome of a random event; can be discrete or continuous.

Expected Value

The long-run average value of a random variable; E(X) = Σ x·P(x) for discrete distributions.

Binomial Distribution

Models number of successes in n independent trials with probability p; P(X=k) = C(n,k)p^k(1-p)^(n-k).

Normal Distribution

A symmetric, bell-shaped continuous distribution defined by mean μ and standard deviation σ.

Z-score

Standardized score z = (x - μ)/σ that tells how many standard deviations x is from the mean.

Variance

A measure of spread calculated as E(X²) - [E(X)]²; standard deviation is the square root of variance.

Flashcards

Probability Mass Function (PMF)

Gives P(X=x) for each value of a discrete random variable; all probabilities sum to 1

Expected value E(X)

Σ x·P(X=x) for discrete; weighted average of possible outcomes

Variance Var(X)

E(X²) - [E(X)]²; measures how spread out the distribution is

Binomial formula

P(X=k) = C(n,k)·p^k·(1-p)^(n-k)

Binomial mean and variance

μ = np; σ² = np(1-p)

Normal distribution

Bell-shaped, symmetric about μ; defined by μ and σ; total area under curve = 1

Empirical (68-95-99.7) rule

68% within 1σ, 95% within 2σ, 99.7% within 3σ of the mean

Z-score

z = (x - μ)/σ; number of standard deviations x is above or below the mean

Questions

1. A fair coin is flipped 8 times. What is the expected number of heads?

   1. 2
   2. 3
   3. 4
   4. 5
   Show answer

   Answer: 4

   E(X) = np = 8 * 0.5 = 4.

2. For a binomial distribution with n = 10 and p = 0.3, the variance is ___.

   Show answer

   Answer: 2.1

   Var(X) = np(1-p) = 10 * 0.3 * 0.7 = 2.1.

3. In a normal distribution, approximately 95% of data falls within one standard deviation of the mean.

   Show answer

   Answer: False

   Approximately 68% falls within 1 standard deviation; 95% falls within 2 standard deviations.

4. A normal distribution has μ = 50 and σ = 10. What is the z-score for x = 65?

   1. 0.5
   2. 1.0
   3. 1.5
   4. 2.0
   Show answer

   Answer: 1.5

   z = (x - μ)/σ = (65 - 50)/10 = 15/10 = 1.5.

5. If a discrete random variable has P(X=1)=0.3, P(X=2)=0.5, P(X=3)=0.2, then E(X) = ___.

   Show answer

   Answer: 1.9

   E(X) = 1(0.3) + 2(0.5) + 3(0.2) = 0.3 + 1.0 + 0.6 = 1.9.

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