23.3 Random Variables & Probability Distributions

Why Named Distributions Matter on the PA-CAT

The PA-CAT Statistics blueprint (Bulletin Table 11, PA-CAT Bulletin of Information, rev. 20240815) groups probability distributions under Anticipating Patterns. You will not be asked to evaluate a complex integral, but you will be asked to choose the correct model for a clinical scenario, compute an expected count, or read a z-table. The skill is model recognition: match the story to the distribution.

Binomial Distribution

A random variable is binomial when four conditions hold:

1. A fixed number n of trials.
2. Each trial has two outcomes, "success" and "failure."
3. The probability of success p is constant across trials.
4. Trials are independent.

The probability of exactly k successes is P(X = k) = C(n,k)·p^k·(1−p)^(n−k), where C(n,k) = n! / [k!(n−k)!]. The mean is μ = np and the variance is σ² = np(1−p).

Clinical Example

A surgeon has a 90% success rate on a particular procedure. In n = 10 independent cases, the expected number of successes is np = 9 and the SD is √(10·0.9·0.1) = √0.9 ≈ 0.95. The probability of exactly 8 successes is C(10,8)·0.9⁸·0.1² ≈ 45 × 0.4305 × 0.01 ≈ 0.194. Watch for the binomial setting trap: if you sample without replacement from a small group (e.g., 10 of 20 nurses), independence fails and you have a hypergeometric situation, not binomial. The PA-CAT rarely tests hypergeometric computation, but it does test whether you notice the violation.

Geometric Distribution

The geometric distribution counts the number of trials until the first success, again with constant p and independent trials. P(X = k) = (1−p)^(k−1)·p, and the mean is 1/p. It is memoryless: the probability of success on the next trial is still p, no matter how many failures preceded it.

Clinical Example

A donor has a 20% chance of matching a recipient. The expected number of donors tested until the first match is 1/0.20 = 5. The probability that the first match is on the third donor is 0.8²·0.2 = 0.128.

Poisson Distribution

The Poisson distribution models the count of rare events in a fixed interval of time, length, area, or volume when events occur independently at a constant rate λ. P(X = k) = e^(−λ)·λ^k / k!, with mean and variance both equal to λ. A useful Poisson fact: if events occur at rate λ per hour, the count in t hours is Poisson(λt).

Clinical Example

An ED averages λ = 4 trauma arrivals per hour. The probability of exactly 2 arrivals in the next hour is e^(−4)·4²/2! ≈ 0.0183 × 16 / 2 ≈ 0.147. In a 3-hour window the rate becomes 12, so the expected count is 12. The Poisson is the natural model for rare adverse events, medication errors per 1,000 doses, or new cancers per person-year.

Normal Distribution

The normal distribution is the continuous bell curve defined by mean μ and SD σ, written N(μ, σ²). Its density is symmetric about μ, with inflection points at μ ± σ. The empirical (68–95–99.7) rule states:

• About 68% of values fall within μ ± σ.
• About 95% fall within μ ± 2σ.
• About 99.7% fall within μ ± 3σ.

z-Scores and the Standard Normal

To use normal tables, convert any normal value to a z-score: z = (x − μ) / σ. The standard normal is N(0, 1). If serum sodium is N(140, 4²), a value of 148 corresponds to z = (148 − 140)/4 = +2, which is at about the 97.5th percentile. A z of −1.96 and +1.96 capture the middle 95% — values you will meet again in confidence intervals.

Checking Normality

Not every clinical variable is normal. Serum sodium and birth weight are approximately normal; hospital charges and lengths of stay are not. A normal probability plot that is roughly linear supports normality; systematic curvature signals skew. When n is large the Central Limit Theorem rescues inference on means even from non-normal parents, but it does not rescue inference on individual extreme values.

Sampling Distributions and the Central Limit Theorem

A sampling distribution is the distribution of a statistic (like x̄) over all possible samples of size n. The single most important result in introductory statistics is the Central Limit Theorem (CLT): as n grows, the sampling distribution of the sample mean x̄ approaches a normal distribution with mean μ and SD σ/√n, regardless of the shape of the parent population.

The SD of the sampling distribution is called the standard error of the mean, SE(x̄) = σ/√n. Three practical consequences:

1. Averages are more stable than individuals. Averaging 25 glucose readings cuts variability by a factor of 5 (√25).
2. Normality is free for large n. Even a strongly skewed population yields a roughly normal x̄ when n is large enough; n ≥ 30 is the conventional rule of thumb.
3. Sample size, not population size, drives precision. Sampling 100 from a city of 100,000 gives nearly the same precision as sampling 100 from a city of 1,000,000.

Worked CLT Example

Suppose patient LOS has mean 4.2 days and SD 3.0 days (right-skewed). For samples of n = 36, the sampling distribution of x̄ is approximately normal with mean 4.2 and SE = 3.0/√36 = 0.5 days. The probability that the sample mean exceeds 5.0 days is P(Z > (5.0 − 4.2)/0.5) = P(Z > 1.6) ≈ 0.055. Notice that the individual LOS distribution was skewed, yet we used the normal model on x̄ — that is the power of the CLT.

Law of Large Numbers

The law of large numbers (LLN) says that as n grows, the sample mean x̄ converges to the population mean μ. This is different from the CLT: the LLN is about where x̄ lands (on μ), while the CLT is about the shape of the distribution of x̄ (normal). The LLN justifies why larger studies give more stable estimates and why casino odds hold over the long run.

Choosing the Right Distribution on the PA-CAT

A quick check: "number of successes in n trials" → binomial; "number of trials until first success" → geometric; "number of events in an interval" → Poisson; "a continuous measurement with known μ, σ" → normal. These phrase-level matches resolve most PA-CAT items in seconds.

Combining Independent Random Variables

The Bulletin's Anticipating Patterns group lists Combining Independent Random Variables as its own objective, and it is the one rule candidates most often get backwards. Means always add. Standard deviations never add — variances do, and only when the variables are independent.

The Four Rules

For any random variables X and Y and constants a and b:

1. E(aX + b) = a·E(X) + b — a linear rescale moves the mean the same way it moves the data.
2. Var(aX + b) = a²·Var(X), so SD(aX + b) = |a|·SD(X). Adding a constant shifts the distribution without changing its spread; multiplying scales the SD by |a|.
3. E(X ± Y) = E(X) ± E(Y) — always true, independent or not.
4. Var(X ± Y) = Var(X) + Var(Y) when X and Y are independent — note the plus sign in both cases. Subtracting two independent variables makes the result more variable, not less, because two sources of randomness still combine.

Rule 4 is the trap. If X and Y are not independent, the correct expression is Var(X ± Y) = Var(X) + Var(Y) ± 2·Cov(X, Y), and the PA-CAT signals this by describing measurements taken on the same subject.

Worked Example 1 — Summing Independent Measurements

A clinic's morning session has a mean of 12 patients with SD 3, and its afternoon session has a mean of 18 patients with SD 4; the sessions are staffed separately and are independent. For the full day:

• Mean = 12 + 18 = 30 patients.
• Variance = 3² + 4² = 9 + 16 = 25, so SD = √25 = 5 patients.

Notice that the SD is 5, not 3 + 4 = 7. Adding SDs directly is the single most common error on this objective.

Worked Example 2 — Differences and Rescaling

A lab reports serum sodium twice on the same specimen using two independent analyzers, each unbiased with SD 1.2 mmol/L. The difference between the two readings has mean 0 and variance 1.2² + 1.2² = 2.88, so SD = 1.70 mmol/L — larger than either instrument alone. If a technician then reports the average of the two readings, that is 0.5(X + Y): its variance is 0.5²(1.44 + 1.44) = 0.72 and its SD is 0.85 mmol/L, which is 1.2/√2. Averaging independent measurements reduces variability by √n; differencing them increases it.

Worked Example 3 — Unit Conversion

Body temperature in a sample has mean 37.0 °C and SD 0.40 °C. Converting to Fahrenheit uses F = 1.8C + 32, so the mean becomes 1.8(37.0) + 32 = 98.6 °F and the SD becomes 1.8(0.40) = 0.72 °F. The +32 shifts the mean but leaves the SD untouched — a favorite PA-CAT distractor adds 32 to the standard deviation as well.

Independent Sums That Stay in the Same Family

Two results worth memorizing because they save time: the sum of independent normal variables is normal (with the mean and variance added as above), and the sum of independent Poisson variables is Poisson with rate λ₁ + λ₂. This is why an ED averaging 4 traumas per hour has a Poisson(12) count over three hours, and why the sample mean of normal data is itself exactly normal at any n.