MH Board · Chapter

Probability Distributions

The concepts MH Board needs, one connected idea at a time.

0 / 7

concepts mastered

Discrete random variables0/4
Continuous random variables0/3

Ready for you

Up nextRandom variablesA random variable gives a number to each outcome of an experiment; it is discrete when its values can be listed, continuous when they fill an intervalStart

Whole chapter

How the ideas build on each other. Each one opens once you've mastered what it needs.

Discrete random variables0 / 4

Random variablesA random variable gives a number to each outcome of an experiment; it is discrete when its values can be listed, continuous when they fill an intervalStart
Probability mass functionA p.m.f. lists $P(X = x)$ for each value: every probability is between $0$ and $1$

Continuous random variables0 / 3

Probability density functionA p.d.f. has $f(x) \ge 0$ and $\int f(x)\,dx = 1$

and they add to

1

;

use the total to find an unknown

k

Locked

Writing a probability distributionList every value the random variable can take, find the probability of each from the experiment, and check that they add to 1Locked

Expected value and variance

E(X) = \sum x\,P(x)

;

\text{Var}(X) = \sum x^2P(x) - [E(X)]^2

; the standard deviation is

\sqrt{\text{Var}(X)}

Locked

over its range;

P(a < X < b) = \int_a^b f(x)\,dx

,

the area under the curve

Locked

Cumulative distribution function

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

: a running total of the p.m.f. for a discrete variable,

\int_{-\infty}^{x} f(t)\,dt

for a continuous oneLocked

Expected value of a continuous random variable

E(X) = \int x\,f(x)\,dx

and

\text{Var}(X) = \int x^2 f(x)\,dx - [E(X)]^2

, with integrals in place of the sumsLocked