Cheatsheet: Random variables and expected value

The one idea

A random variable is a number whose value comes from chance. Its expected value is the long-run, probability-weighted average, the backbone of machine-learning objectives.

Definitions

Random variable: a number set by a random outcome.
  Discrete  = listable values (die roll, counts).   Continuous = a range (heights, times).
Probability distribution: each possible value paired with its probability (probabilities sum to 1).

Expected value

E[X] = sum over all values of  (value x its probability)   = long-run average

Fair die:  E[X] = (1+2+3+4+5+6)/6 = 21/6 = 3.5   (never an actual roll: it's an average)
Decision:  win $10 w.p. 0.2, lose $3 w.p. 0.8
           E = (+10)(0.2) + (-3)(0.8) = 2 - 2.4 = -0.40 per play   (long-run loser)

The expected value need not be an achievable outcome. Weight by probability (not a plain average unless values are equally likely).

Variance and standard deviation

Var(X) = sum over all values of  (value - E[X])^2 x its probability
Std    = square root of Var(X)   (back in original units)

$1 coin bet (+1 / -1, each 0.5):
  E[X]=0,  Var = (1)^2(0.5)+(-1)^2(0.5) = 1,  Std = 1

Same expected value + more variance = more risk in any single outcome.

In machine learning

Phrase	Means
”Minimize the loss”	Make the expected error (a random variable) small
”Maximize reward”	Maximize expected total reward; pick actions by expected value
”95% accurate”	An expected value over the data distribution

Pitfalls to dodge

• Treating the expected value as the most likely or guaranteed value (it is a long-run average).
• Ignoring variance (same expected value can mean very different risk).
• Expecting the average to appear in a few trials (it is a long-run idea).
• Forgetting to weight by probability (not a plain average unless equally likely).

Words to use precisely

• Random variable: a number whose value is a random outcome.
• Expected value E[X]: the probability-weighted average; the long-run mean.
• Variance: probability-weighted average squared distance from E[X].
• Standard deviation: square root of variance; spread in original units.