Random Variables and Probability Distributions

Definition and Types of Random Variables

A random variable is a numeric value associated with the outcome of a probability experiment. Random variables are classified as either discrete or continuous:

• Discrete Random Variable: Takes on a countable set of values, often whole numbers. Typically associated with counting (e.g., number of defective items).

• Continuous Random Variable: Can take on any value within an interval. Typically associated with measurement (e.g., weight, time, volume).

Common notation for random variables includes X, Y, and Z.

Probability Distributions

• Discrete Probability Distribution: Represented by a table or chart listing all possible values and their probabilities.

• Continuous Probability Distribution: Represented by a smooth curve called a probability density function (PDF). The total area under the curve equals 1.

Key Properties and Formulas for Discrete Random Variables

• Mean (Expected Value):

• Variance:

Key Discrete Random Variables

• Binomial Random Variable:

  • Fixed number of independent trials

  • Each trial has two outcomes: success or failure

  • Constant probability of success, p

  • Counts the number of successes in n trials

  • Calculator commands: BINOMPDF (exactly X successes), BINOMCDF (X or fewer successes)

• Poisson Random Variable:

  • Counts the number of events in a fixed interval of time or space

  • Events occur independently

  • Constant average rate, λ (lambda)

  • Calculator commands: POISSONPDF (exactly X events), POISSONCDF (X or fewer events)

• Hypergeometric Random Variable:

  • Sampling without replacement from a finite population with known composition

  • Useful for probability of a certain composition in the sample (e.g., selecting colored marbles from a jar)

Continuous Random Variables

• Result from measurement; can take any value within an interval

• Probability distribution described by a probability density function (PDF)

• Probability is the area under the PDF curve over an interval

Normal Random Variable

• Most important continuous random variable

• PDF is bell-shaped, centered at the mean (μ), with inflection points at one standard deviation (σ) from the mean

• Calculator commands: NORMALCDF (probabilities), INVNORM (find value for a given percentile)

Standard Normal Random Variable (Z)

• Mean = 0, Standard deviation = 1

• Z-scores represent the number of standard deviations from the mean

• Useful for comparing different normal distributions (e.g., SAT vs. ACT scores)

Sampling Distributions

Definition and Importance

A sampling distribution is the probability distribution of a statistic (such as the sample mean or sample proportion) computed from a random sample. Sampling distributions help us understand the variability of sample statistics and make inferences about population parameters.

Key Properties

• Sample averages (\bar{x}) and sample proportions (\hat{p}) are continuous random variables.

• Their probability distributions are called sampling distributions.

Formulas for Sampling Distributions

• Sample Mean:

  • Expected value:

  • Standard deviation: (approximated by for large samples)

• Sample Proportion:

  • Expected value:

  • Standard deviation:

Central Limit Theorem (CLT)

• If sample size n ≥ 30, the sampling distribution of the sample mean is approximately normal, regardless of the population's distribution.

• If the population is normal, the sampling distribution of the sample mean is normal for any sample size.

• Larger samples yield a tighter (less variable) sampling distribution.

Estimator Properties

• Unbiased Estimator: An estimator whose expected value equals the population parameter (e.g., sample mean for population mean).

• Minimum Variance: Among all unbiased estimators, the one with the smallest variance is preferred.

Confidence Intervals for Population Mean (μ) and Proportion (p)

Definition and Interpretation

A confidence interval (CI) is a range of values, derived from sample statistics, that is likely to contain the population parameter with a specified level of confidence (e.g., 95%).

• Confidence Level (CL): The probability that the CI contains the parameter in repeated sampling (e.g., 95%).

• Margin of Error: The half-width of the CI; reflects sampling variability.

Constructing Confidence Intervals

• Large Sample CI for μ: Use the normal distribution (Central Limit Theorem applies). Substitute sample standard deviation s for population σ if unknown. Calculator command: Zinterval Critical value: Find z for area α/2 in each tail using INVNORM.

• Small Sample CI for μ: Use the t-distribution if the population is approximately normal. Calculator command: Tinterval Degrees of freedom (DF): n – 1 Critical value: Find t for area α/2 in each tail using INVT.

• Large Sample CI for p: Requires at least 15 successes and 15 failures in the sample. Use normal approximation. Calculator command: 1prop-ZINT

• Small Sample CI for p: Used when there are fewer than 15 successes or failures. No standard calculator command; see detailed notes for methods.

• Determining Sample Size: To achieve a desired margin of error, use p = 0.5 if no prior estimate is available.

• Alpha (α): α = 1 – confidence level; determines critical z or t values (area α/2 in each tail).

Hypothesis Testing for Population Mean (μ) and Proportion (p)

Key Concepts and Steps

1. Null Hypothesis (H₀): Represents the status quo; always contains an equality (e.g., H₀: μ = μ₀).

2. Alternative Hypothesis (Hₐ): Represents the claim to be tested; uses >, <, or ≠.

3. Type I Error (α): Rejecting H₀ when it is true.

4. Type II Error (β): Failing to reject H₀ when it is false.

5. Significance Level (α): Probability of a Type I error; sets the threshold for rejection.

6. Test Statistic: Converts the sample statistic to a z or t value for comparison.

7. Critical Value: The z or t value marking the start of the rejection region.

8. Rejection Region: Area(s) in the tails where H₀ is rejected.

9. P-value: Probability of observing a result as extreme as the sample, assuming H₀ is true.

10. Decision Rule:

    • If p-value < α, reject H₀.

    • If p-value > α, fail to reject H₀ (H₀ is plausible).

    • Alternatively, compare test statistic to critical value(s).

11. Reporting Results:

    • "There is sufficient evidence at α = xx to reject H₀ and accept Hₐ."

    • "There is insufficient evidence at α = xx to reject H₀. H₀ is plausible."

Types of Tests

• One-Tailed Test (Left): Hₐ: parameter < value

• One-Tailed Test (Right): Hₐ: parameter > value

• Two-Tailed Test: Hₐ: parameter ≠ value

Test Selection and Calculator Commands

• Large Sample Test for μ: Use normal distribution; ZTEST

• Small Sample Test for μ: Use t-distribution if population is normal; TTEST

• Large Sample Test for p: Use normal distribution if n*p₀ and n*(1–p₀) ≥ 15; 1propZtest

Summary Table: Key Random Variables and Their Properties

Example: Constructing a 95% Confidence Interval for the Mean

• Suppose a sample of n = 36 has a mean of 50 and a standard deviation of 12.

• 95% CI for μ:

• For 95% confidence,

• Margin of error:

• CI: (50 – 3.92, 50 + 3.92) = (46.08, 53.92)

Example: Hypothesis Test for a Proportion

• Suppose in a sample of 100, 60 are successes. Test H₀: p = 0.5 vs. Hₐ: p ≠ 0.5 at α = 0.05.

• Test statistic:

• , ,

• p-value ≈ 0.0455 (two-tailed)

• Since p-value < 0.05, reject H₀.

Additional info: Some calculator commands and detailed procedures are referenced for classroom calculators (e.g., TI-84), but the underlying statistical principles apply universally.