Foundations — Uniform, Exponential, and Beta distributions

This page assumes you know nothing beyond arithmetic and reading a graph. We build every tool the parent uses, in the order it needs them, so no symbol appears before you own it.

1. A number that comes out random: the random variable $X$

Picture: think of a dart landing somewhere on a ruler. Before the throw, the landing spot is $X$. After the throw, "it landed at $0.3$" is a value $x=0.3$.

Why the topic needs it. All three distributions describe what number is likely to come out: a random weight, a random waiting time, a random probability. Without $X$ there is nothing to describe.

• $X\sim U(a,b)$ — the little "$\sim$" reads "is distributed as". It says: "the box $X$ follows the recipe on the right." So $X\sim U(a,b)$ = "$X$ is a Uniform box between $a$ and $b$."

2. The blanket of belief: the probability density $f(x)$

For a continuous box (one that can land on any real number, not just whole numbers), asking "what is the chance $X$ equals exactly $0.3000\dots$?" is hopeless — there are infinitely many points, so any single one has chance $0$. Instead we measure how densely packed the belief is near each point.

Look at the figure: the shaded amber region between $x=1$ and $x=2$ is the probability that $X$ lands in $[1,2]$. Its area — not the curve's height — is the number between $0$ and $1$.

See Continuous Random Variables and PDFs for the full treatment — here we only need the picture.

Why the topic needs it. Every "PDF formula" box in the parent ($\frac{1}{b-a}$, $\lambda e^{-\lambda t}$, $\frac{1}{B}{x}^{\alpha -1}(1-x{)}^{\beta -1}$) is just the shape of this blanket for that distribution.

3. The tool that measures area: the integral $\int$

The moment we said "probability = area," we needed a machine that computes area under a curved top. That machine is the integral. We chose it — not multiplication — because the blanket's height changes as $x$ moves, so we can't just do "height × width."

• $dx$ = an infinitely thin sliver of width along the ruler.
• ${\int }_a^b$ = "sum the strips from the left edge $a$ to the right edge $b$."
• The bar rule ${\left[g(x)\right]}_a^b=g(b)-g(a)$ means "plug in the top edge, subtract the bottom edge." The parent uses this in every derivation.

Why the topic needs it. The Uniform mean ${\int }_a^{b}x\frac{1}{b-a}dx$, the Beta normaliser ${\int }_0^1{x}^{\alpha -1}(1-x{)}^{\beta -1}dx$, and the check "$\int cdx=1$" are all this same summing machine.

4. The running total from the left: the CDF $F(x)$

Instead of "area over a middle stretch," often we want "area from the far left up to a point $x$" — the total belief that $X$ came out at most $x$.

The staircase-smooth curve in the figure only ever rises. Its height at $x$ equals the shaded area to its left in the PDF picture.

Why the topic needs it. The Exponential's $F(t)=1-e^{-\lambda t}$ and Uniform's $F(x)=\frac{x-a}{b-a}$ are CDFs. Percentile questions ("95th percentile wait time") solve the CDF equation for $x$.

5. The balance point and the spread: $E[X]$ and $\text{Var}(X)$

Why the parent uses $E[X^2]-(E[X]{)}^2$. It's the same variance, rearranged so you only need two integrals ($\int xf$ and $\int x^{2}f$) instead of expanding $(x-\mu {)}^2$ by hand. See Maximum Likelihood Estimation for where means and variances get estimated from data.

6. The number $e$ and the exponential $e^{-\lambda t}$

The Exponential distribution is named after this. We need it before the parent uses $e^{-\lambda t}$.

• $\lambda$ (Greek "lambda") controls how fast it decays: bigger $\lambda$ = steeper drop.
• $e^{-\lambda t}$ answers "what fraction of belief is still to the right of time $t$?" — that's exactly $P(T>t)$, the chance you're still waiting.
• $\ln$ (natural log) is the undo button for $e$: $\ln (e^x)=x$. The parent uses it to solve $e^{-0.5t}=0.05$ for $t$ — take $\ln$ of both sides to bring $t$ down from the exponent.

Why the topic needs it. Every Exponential formula ($f=\lambda e^{-\lambda t}$, $F=1-e^{-\lambda t}$, memorylessness) rides on this decaying curve. It flows from the Poisson Process, which counts events happening at steady rate $\lambda$.

7. Greek parameters: $a,b,\lambda ,\alpha ,\beta$ and the shape knobs

8. The factorial-for-any-number: $\Gamma$ and the Beta function $B$

Why the topic needs it. Beta lives on $[0,1]$ to model a probability of a probability. This is the engine of Bayesian Inference and Conjugate Priors and powers Thompson Sampling.

Prerequisite map

Uniform, Exponential and Beta all feed the topic — and the topic itself feeds Monte Carlo Methods, Xavier and He Initialization, and Bayesian ML downstream.

Equipment checklist

Cover the right side; say your answer aloud; then reveal.