Probability distributions in finance

A distribution is the complete rulebook for a random variable, every value it can take, and how much probability mass sits at (or near) each one. Six distributions carry almost all of applied finance, and each one is either a natural model of some finance situation directly, or a mathematical consequence of the others that makes estimation possible. Learn each one's shape, parameters, mean and variance, and the one finance scenario it is the obvious tool for, that quartet is worth more than memorizing any single formula.

What is the binomial distribution?

Binomial: counting successes in independent trials. A single yes/no trial with success probability p is a Bernoulli trial. Run n independent, identical Bernoulli trials and count the successes, and that count X follows a Binomial(n, p) distribution:

P(X = k) = C(n, k) · p^k · (1 − p)^(n − k)          E[X] = np          Var(X) = np(1 − p)

where C(n, k) = n! / [k!(n − k)!] counts the number of ways to choose which k of the n trials were successes. The binomial applies exactly when three conditions hold: a fixed number of trials, each trial independent of the others, and a constant success probability across trials, a portfolio of bonds each defaulting independently with the same probability, an IPO allotment lottery, a fixed batch of loan applications each approved or declined. When n is reasonably large, the binomial is well approximated by a normal distribution with the same mean and variance (np and np(1 − p)), a special case of the Central Limit Theorem, and the standard rule of thumb for a trustworthy approximation is np ≥ 5 and n(1 − p) ≥ 5. Because you are approximating a discrete count with a continuous curve, apply a continuity correction: to approximate P(X ≥ 10), use P(Z ≥ (9.5 − mean)/sd), shifting the boundary half a unit to account for the mismatch. Worked Example 2 runs this exact comparison on a bond-default count and shows how close (and how imperfect) the approximation is.

What is the normal distribution?

Normal, the workhorse. The familiar symmetric bell curve, fully described by its mean μ and variance σ². Any normal variable can be standardized into a z-score, z = (x − μ)/σ, which tells you how many standard deviations x sits from the mean and lets you read probabilities off one universal standard-normal table (or one scipy.stats.norm call) regardless of the original units. The empirical rule is worth holding as a fast mental check: about 68.3% of a normal distribution's mass sits within ±1 standard deviation of the mean, 95.4% within ±2, and 99.7% within ±3 (verified: Φ(1) − Φ(−1) = 68.27%, Φ(2) − Φ(−2) = 95.45%, Φ(3) − Φ(−3) = 99.73%). The normal distribution is ubiquitous in finance for a reason the Central Limit Theorem makes precise, sums and averages of many small, roughly independent effects tend toward it, and it is the standard first approximation for asset returns and for the tail behavior of screening scores. On that last point: M2.07's Beneish M-Score is a probit score, and its published flag threshold of M > −1.78 is, in exactly this module's language, the statement Φ(−1.78) ≈ 3.75% (M2.07 rounds this to "≈3.8%", consistent), a company clearing the threshold resembles a population where only about 1 in 26 clean firms would score this high by chance. The wider screening threshold M2.07 also mentions, −2.22, corresponds to Φ(−2.22) ≈ 1.32% ("≈1.3%" there). And the specific synthetic company M2.07 scored, at M = −1.42, sits at Φ(−1.42) ≈ 7.78% under this same reading: comfortably above the 3.75% baseline, which is exactly why it was flagged. None of the Beneish indices are recomputed here (that is M2.06/M2.07's construction, never repeated in this module); this is only the CDF machinery that makes an already-published score legible in probability terms. One caveat stated prominently, not buried: real asset returns are not exactly normal, they typically show fatter tails (more extreme moves than a true normal would predict) and often negative skew (crashes are sharper than rallies). The normal model here is a disciplined, tractable starting approximation, not a law of nature, and this Ring's later risk modules (AL1.03 and others) will lean on that exact caveat when it matters.

What is the lognormal distribution?

Lognormal, when the normal variable is a logarithm. If X = ln(Y) follows a normal distribution N(μ, σ²), then Y itself is called lognormal. This single fact solves a real problem: a normal variable can be negative, but a price level never can be, so prices are not modeled as normal directly, their continuously compounded returns (log-returns) are modeled as normal instead, and the price itself falls out as lognormal. Two facts about a lognormal Y follow from the underlying normal parameters:

median(Y) = e^μ          mean(Y) = e^(μ + σ²/2)

Notice the mean exceeds the median, a direct consequence of the distribution's right skew (a lognormal has a long right tail, since Y = e^X can grow without bound but is floored at zero). Why does this matter beyond the algebra? Because a stock's log-return over a period is, to a first approximation, a sum of many smaller shocks, and sums of many roughly-independent shocks tend toward normal (the CLT again, arriving early), so the log-price change tends toward normal, and the price level itself tends toward lognormal. This is the entire justification for the shape of the Black-Scholes-Merton option-pricing formula; DV1.02 is where that formula, its assumptions, and its Greeks are built and is not previewed or derived here, this module's job stops at the distribution itself, which Worked Example 3 puts real numbers through.

When do you use the t-distribution?

t-distribution, when you don't actually know σ. Every confidence interval and test statistic built on a standardized sample mean needs to divide by a standard deviation, but in practice, the population standard deviation σ is essentially never known; you only have the sample standard deviation s, itself an estimate with its own error. Standardizing with s instead of the true σ, especially with a small sample, produces a distribution with the same bell shape as the normal but fatter tails, the t-distribution, indexed by a single parameter, degrees of freedom, df = n − 1 for a sample of size n. The fatter tails are not a technicality; they are the honest price of not knowing σ, an extra margin of uncertainty baked directly into the shape of the distribution. As df grows, that extra uncertainty shrinks and the t-distribution converges to the standard normal, concretely, the two-sided 95% critical value falls from 2.5706 at df = 5, to 2.2281 at df = 10, 2.0860 at df = 20, 2.0423 at df = 30, 2.0003 at df = 60, 1.9840 at df = 100, and 1.9623 at df = 1,000, closing in on the normal's familiar 1.96 (all verified by direct computation). This is the distribution behind every small-sample confidence interval for a mean, including Worked Example 6.

What are the chi-square and F distributions?

Chi-square, the distribution of a sum of squared normals. Take k independent, standard-normal variables Z₁, …, Zₖ, square each one, and add them up: Z₁² + Z₂² + … + Zₖ² follows a chi-square distribution with k degrees of freedom. It is always positive (you summed squares), right-skewed for small k, and looks progressively more normal-shaped as k grows, itself a small echo of the Central Limit Theorem, since a chi-square variable is, after all, a sum. Its mean equals its degrees of freedom, and its variance equals twice its degrees of freedom (verified: for df = 24, mean = 24.0 and variance = 48.0 = 2×24). The fact that makes chi-square directly useful for estimation (this module's business) is that a rescaled sample variance follows this distribution exactly:

(n − 1)·s² / σ²  ~  chi-square with (n − 1) degrees of freedom

which is precisely the fact a confidence interval for a population variance is built from (Worked Example 7). Note what this module deliberately does not do with chi-square: the formal chi-square test (a stated significance level, a decision rule, a goodness-of-fit or independence conclusion) is QM1.02's territory. Here, chi-square is a distribution you locate a statistic inside of for the purpose of estimation, nothing more.

F-distribution, the ratio of two (scaled) chi-squares. If U follows a chi-square distribution with d₁ degrees of freedom and V follows one with d₂ degrees of freedom, and U and V are independent, then the ratio

F = (U / d1) / (V / d2)   ~   F-distribution with (d1, d2) degrees of freedom

is always positive and right-skewed, with a mean of d₂/(d₂ − 2) for d₂ > 2, for an F(24, 30), that mean works out to 1.0714, close to but not exactly 1 (verified directly). The F-distribution arises naturally when you compare the spreads of two independent samples, because each sample's rescaled variance is itself connected to a chi-square distribution, so the ratio of two sample variances, under standard assumptions, follows an F-distribution with numerator and denominator degrees of freedom equal to (n₁ − 1) and (n₂ − 1). Worked Example 7 computes exactly this ratio for two funds' return variances and locates it inside its F-distribution. As with chi-square, the boundary matters: this module computes the ratio and reads off where it sits; turning that into a formal decision (a stated significance level, a reject/fail-to-reject conclusion) belongs to QM1.02's two-variance test (and, in a different guise, to its multiple-regression F-test for overall significance), and is not conducted here.