Bring Statistics and Probability Together

Probability starts with a model and predicts possible data. Statistics starts with observed data and learns about the model. The random variable and its distribution are the shared language between them.

An experiment produces an outcome. A random variable assigns a number to that outcome. A probability distribution assigns probabilities to the possible numbers. Statistics summarizes or learns from realized values.

Yes. From the same two-dice experiment, X can be the sum, Y an indicator that the sum is seven, and Z the maximum die. The experiment is unchanged; the numerical question changes.

For every possible value x, collect the experiment outcomes that make X equal x and add their probabilities. This pushforward turns an outcome model into the probability distribution of X.

The probability mass function p(x)=P(X=x) lists the probability of each possible value. Every mass is nonnegative and all masses sum to one.

The expected value multiplies each possible value by its probability and adds the products. It is the distribution's probability-weighted balance point, not a guarantee for the next observation.

Two distributions can have the same expected value while concentrating values very differently. Center answers where the distribution balances; spread answers how far values range around it.

The population mean μ is a fixed parameter, not a random variable. Therefore μ² is constant, and the expectation of a constant is the constant itself.

Mean is the balance point, median splits probability in half, and mode is a most probable value. They may coincide in a symmetric unimodal distribution but can separate when shape is skewed or multimodal.

A value m is a median when at least half the probability is at or below m and at least half is at or above m. Both inequalities matter, and a distribution can have more than one median.

A mode is any value that maximizes the PMF. One distribution can be unimodal, bimodal, or have several modes; the mode need not equal the mean or median.

Because E[X-μ]=0, positive and negative deviations cancel exactly. Absolute or squared deviations prevent cancellation; variance uses squares for smooth algebra and strong theoretical properties.

Variance averages squared distances from the mean, weighted by their probabilities. Squaring makes every contribution nonnegative and gives larger deviations greater influence.

Variance is measured in squared units. Its square root, the standard deviation, returns to the original units and describes a typical scale of distance from the mean without fixing an exact range.

Expanding (X-μ)² and using linearity gives Var(X)=E[(X-μ)²]=E[X²]-μ². The first formula explains distance from the mean; the second is often faster to compute.

Expand the square, take expectation term by term, replace E[X] with μ, and treat μ as constant. The cancellation leaves E[X²]-μ².

Define a new random variable D=(X-μ)². Then Var(X)=E[D]. The same probability-weighted averaging rule applies; only the quantity being averaged has changed.

If the probability distribution is known, mean, median, mode, and variance can be computed directly from it. With sample data, statistics estimate corresponding population features and include sampling uncertainty.

Choose a variable, specify or learn its distribution, compute interpretable summaries, compare those summaries with observed data, quantify sampling uncertainty, and answer the original question without confusing a model feature with a sample estimate.