Random variables and their distributions

An outcome like "the coin came up heads" is hard to integrate. A number like "the number of heads in ten tosses" is not. A random variable is the bridge: a function that reads an outcome and returns a number, so that probability becomes a distribution over the number line.

Switch between the binomial, Poisson and normal laws below. The bars or curve are the pmf or pdf; the shaded area is the probability up to the threshold.

Discrete and continuous #

A discrete random variable takes a countable set of values. Its probability mass function (pmf) gives $p(k) = P(X = k)$, with

$\sum_k p(k) = 1.$

A continuous random variable takes values in an interval. Its probability density function (pdf) $f(x)$ satisfies $f(x) \ge 0$ and

$\int_{-\infty}^{\infty} f(x)\,dx = 1.$

For continuous variables, $P(X = x) = 0$ for every single point; only intervals have non-zero probability, given by the integral. This is the single biggest conceptual break from the discrete case, and it is why the normal distribution is drawn as a density rather than a set of bars.

Both kinds share the cumulative distribution function (cdf)

$F(x) = P(X \le x),$

which is non-decreasing, goes from 0 to 1, and works identically for both. The widget's threshold readout is a cdf value.

The named distributions #

Three laws cover an enormous amount of ground:

• Binomial. $X$ is the number of successes in $n$ independent trials with success probability $p$. The widget computes $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$, with mean $np$ and variance $np(1-p)$. The shape is symmetric when $p = 0.5$ and skews as $p$ moves to the extremes.
• Poisson. $X$ counts events in a fixed interval when they occur independently at constant rate $\lambda$. Then $P(X=k) = e^{-\lambda}\lambda^k / k!$, and — the famous fact — mean and variance are both $\lambda$. It is the limit of the binomial when $n$ grows and $np \to \lambda$.
• Normal. The bell curve, with density $f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-(x-\mu)^2 / 2\sigma^2}$, mean $\mu$ and variance $\sigma^2$. It is the limit in the central limit theorem, which is why it appears whenever many small independent effects are added.

Standardization #

Subtracting the mean and dividing by the standard deviation gives a standardized variable with mean 0 and variance 1:

$Z = \frac{X - \mu}{\sigma}.$

This is how a normal table or a computer cdf can serve every normal distribution at once: transform to $Z$, then look up. It is also why standardizing features is a standard first step in machine learning — many estimators behave as if the inputs were already on a common scale.

Check yourself

Eduspheria wiki · Mathematics for AI, Probability

0 / 5 answered

1. 1X is Binomial(n = 20, p = 0.3). What is E[X]?

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2. 2X is Poisson with λ = 4. What is P(X = 0)? Give four decimals.

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3. 3Which function gives P(X ≤ x) for both discrete and continuous random variables?

   A.The probability mass functionB.The probability density functionC.The likelihood functionD.The cumulative distribution function

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4. 4For a continuous random variable, P(X = 5) = 0.

   TrueFalse

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5. 5What is the operation called that subtracts the mean and divides by the standard deviation?

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Where next: expectation and variance — the two numbers that summarize a distribution.