Probability Distributions Exercises in R: 25 Real-World Problems

Explanation: pnorm(q) returns the lower-tail CDF P(Z <= q) by default. For the upper tail you can either subtract from 1 or pass lower.tail = FALSE, which is more numerically stable in the deep tail. The "within one standard deviation" probability 0.6827 is the famous 68 from the 68/95/99.7 rule, computed as the difference of two CDFs.

Explanation: pnorm(130, 100, 15, lower.tail = FALSE) returns P(X > 130) directly without subtracting from 1. About 2.3 percent of adults score above 130, which matches the well-known "two standard deviations above the mean" rule of thumb. Avoid 1 - pnorm(130, 100, 15) for very extreme values because of floating-point cancellation; lower.tail = FALSE evaluates the survival function in log space internally.

Explanation: The reject region is two disjoint tails, so add the lower-tail probability at 492 to the upper-tail probability at 508. Because the spec limits are exactly 2 standard deviations from the mean, the answer should be close to the canonical 4.55 percent outside-2-sigma figure. For symmetric limits a faster equivalent is 2 * pnorm(-2); this code generalizes to asymmetric tolerances.

Explanation: qnorm(p) inverts pnorm(): it returns the value x such that P(X <= x) = p. The 90th-percentile z-score 1.2816 multiplied by sd 200 plus mean 1050 gives 1306.3, which rounds to 1306. The d/p/q/r naming pattern is consistent across distributions (qbinom, qpois, qt, etc.), which lets you swap families without re-learning the API.

Explanation: rnorm(n, mean, sd) generates n pseudo-random draws. With n = 1000 the sample mean and sd land within roughly 1 percent of the true parameters, which gives you a tangible feel for sampling variability. The set.seed() call makes the draws reproducible across runs; without it every run yields different numbers, which is fine for production simulation but bad for shared exercises.

Explanation: dbinom(x, size, prob) evaluates the binomial PMF P(X = x) for n trials with success probability p. Roughly 25.5 percent of samples of 50 wafers will contain exactly 2 defectives. A common mistake is passing dbinom(0.03, 50, 2): arguments are positional and the first slot is the count, not the probability. When in doubt name the arguments.

Explanation: P(X >= 8) = P(X > 7) = pbinom(7, lower.tail = FALSE). The lower-tail flag is preferred over 1 - pbinom(7, ...) for numerical accuracy in the tail. Note the off-by-one: pbinom(8, ..., lower.tail = FALSE) would return P(X > 8) = P(X >= 9), which excludes the case X = 8. Always read the inequality direction carefully when using discrete CDFs.

Explanation: The Poisson distribution has a single parameter lambda equal to both the mean and the variance. dpois(x, lambda) evaluates P(X = x). The Poisson is the natural model for counts of independent events in fixed time or space windows: call arrivals, photon detections, web hits, mutations along a DNA segment. If your data has variance much larger than the mean, switch to a negative binomial instead.

Explanation: qpois(p, lambda) returns the smallest integer x such that P(X <= x) >= p. Staffing for 21 simultaneous calls covers 99 percent of hours given lambda = 12. The discrete quantile function steps in integers, so qpois(0.99, 12) and qpois(0.995, 12) may return the same value. For continuous distributions like the normal the quantile is unique; for discrete ones the right-continuous convention is enforced.

Explanation: rbinom(n, size = 1, prob) draws n Bernoulli trials. The empirical proportion 0.3677 differs from the true 0.37 by about 0.0023, which is consistent with the theoretical standard error sqrt(p(1-p)/n) = sqrt(0.370.63/10000) = 0.00483. By the law of large numbers, p_hat converges to p as n grows, but at the slow sqrt(n) rate: to halve the error you need four times as many trials.

Explanation: For Uniform(a, b) the CDF is linear: P(X <= x) = (x - a) / (b - a). The 10-minute window from 20 to 30 spans one-third of the 30-minute support, hence p = 1/3. The theoretical mean is the midpoint (a + b) / 2 = 30. The uniform is a useful baseline for "completely uninformed" probabilities and a building block for inverse-CDF sampling from any other distribution.

Explanation: pexp() uses the rate parameter (reciprocal of the mean), not the mean. With rate = 1/200 the mean is 200 and the upper-tail probability at 500 is exp(-500/200) = exp(-2.5) = 0.0821. About 8 percent of requests will breach the 500 ms threshold. The exponential is memoryless, which makes it the canonical model for waiting times in a Poisson process; if observed data is heavier-tailed than exponential, consider a Weibull or log-normal model.

Explanation: The sum of k independent Exponential(rate) variables is Gamma(shape = k, rate). With shape 5 and rate 1/30 the mean is 5 * 30 = 150 and the 95th percentile is about 273 seconds. The Gamma generalizes both the exponential (shape = 1) and the chi-square (shape = df/2, rate = 1/2). Be careful with parameterization: R lets you supply either rate or scale (with rate = 1 / scale); supplying both raises an error.

Explanation: Beta(alpha, beta) is the conjugate prior for the binomial likelihood, so the posterior after observing s successes and f failures is Beta(alpha + s, beta + f). The posterior mean is alpha / (alpha + beta). The 2.5th percentile qbeta(0.025, ...) is the lower bound of the equal-tailed 95 percent credible interval. Note this is different from a highest-density interval (HDI) for skewed Beta posteriors; the HDInterval package computes the HDI if you need it.

Explanation: plnorm() takes the parameters of the underlying normal, not the lognormal mean and variance. Internally R evaluates pnorm(log(1.025), meanlog, sdlog, lower.tail = FALSE). About 1.95 percent of days will see a price ratio above 1.025 under this model. Lognormal is a workhorse for multiplicative growth processes (asset prices, biological populations, particle sizes) because the log of the variable is symmetric and well-behaved.

Explanation: For a two-sided test at alpha = 0.05 you split the rejection region into two tails of 0.025 each, so the critical value is the 97.5 percentile of the t distribution with df = 19. The t critical 2.093 is larger than the normal critical 1.96, reflecting the heavier tails at finite df. As df grows the t converges to the standard normal; at df = 30 the critical value is already 2.042, very close to 1.96.

Explanation: A one-sided lower-tail test rejects when the chi-square statistic is too small, which happens when the sample variance is unusually small. qchisq(0.05, df = 24) returns the 5th percentile, 13.85. The mean of a chi-square with df = k is k, so 13.85 sits well below the mean of 24, as expected for a lower-tail critical. For the upper-tail version (testing whether variance exceeds a threshold), use qchisq(0.95, df = 24).

Explanation: The F distribution is the ratio of two scaled chi-square variables and is always right-skewed and non-negative. For ANOVA the numerator df is groups minus 1 and the denominator df is total observations minus groups. The upper-tail 95th percentile 2.77 is the rejection cutoff: an observed F above this value implies between-group variance is significantly larger than within-group variance at alpha = 0.05.

Explanation: Read row by row to see two trends. First, the tails (|x| = 3) are progressively heavier for smaller df: the t with df = 3 has roughly 5 times the normal density at x = 3. Second, as df grows the t density approaches the normal everywhere. At df = 30 the differences are already in the third decimal place. This is the formal statement of the t-to-normal convergence theorem, and the reason "large-sample" z procedures replace t procedures when df is high.

Explanation: This demonstrates the Central Limit Theorem on a skewed parent. The exponential with rate 1/10 has true mean 10 and true sd 10, so the theoretical standard error of the sample mean is 10 / sqrt(25) = 2. The empirical clt_se 2.0252 matches that closely. The sampling distribution of the mean is approximately normal even though the parent is exponential, which is why z-based and t-based methods work for averages of moderately sized samples from non-normal populations.

Explanation: The fraction of uniform points falling inside the quarter circle is the ratio of areas (pi/4) / 1 = pi/4, so multiplying by 4 estimates pi. The error shrinks at rate 1/sqrt(n), which is why getting one more digit of accuracy costs 100 times more samples. This is the canonical Monte Carlo example and a stepping stone to high-dimensional integration where deterministic quadrature becomes infeasible.

Explanation: Parametric VaR under a Gaussian log-return model converts a quantile of the log-return into a simple-return quantile via the exponential map. The 1 percent log-return quantile is qnorm(0.01, 0.0004, 0.011) = -0.02519, and exp(-0.02519) - 1 = -0.02487, so the 1-day 99 percent VaR is about 2.49 percent of position value. Real risk teams typically compute historical VaR (the empirical quantile of realized returns) and validate against parametric VaR; large gaps signal fat tails not captured by the normal model.

Explanation: ecdf() returns a step function that you can evaluate at any point. The largest absolute gap between the empirical and theoretical CDFs is the Kolmogorov-Smirnov test statistic; here all gaps are under 0.01, consistent with a fit-to-the-truth scenario. For n = 500 the KS critical value at alpha = 0.05 is about 0.061, so this sample is far from rejecting normality. Use this technique to spot-check distributional fit anywhere you have raw samples.

Explanation: The z interval mean +/- z_{0.975} * sigma / sqrt(n) has nominal coverage 95 percent. The empirical coverage 0.9534 is within Monte Carlo error of the nominal value (the Monte Carlo standard error here is sqrt(0.95 * 0.05 / 5000) = 0.003). If you replace the true sd with the sample sd and keep using qnorm(0.975), coverage falls below nominal for small n, which is the original motivation for the t interval.

Explanation: Plotting two densities on a shared axis is the clearest way to internalize the "heavier tails, lower peak" relationship between t and normal. The t with df = 3 has only one finite variance moment (variance df / (df - 2) = 3) and very heavy tails, which is why t-based inference is so robust to mild outliers. As df increases the t curve squashes toward the normal; at df = 30 they overlap almost exactly.