Probability & Statistics Reference

Definitions

Probability distribution

A rule assigning probabilities to values of a random variable.

Example: A fair die assigns probability

\frac{1}{6}

to each result.

Expected value

E (X) = \sum x p (x)

The probability-weighted long-run average of a random variable.

Example: A fair die has expected value 3.5.

Variance

Var (X) = E [(X - μ)^{2}]

The expected squared distance from the mean.

Example: A larger variance means greater spread.

Correlation

- 1 \leq r \leq 1

A standardized measure of linear association between two quantitative variables.

Example:

r

near

1

indicates a strong positive linear association.

Regression

A model describing how an outcome changes with one or more explanatory variables.

Example: A least-squares line predicts

y

from

x

Z-score

z = (x - μ) / σ

The number of standard deviations a value lies above or below the mean.

Example:

z = - 2

is two standard deviations below the mean.

Normal distribution

A symmetric bell-shaped continuous distribution determined by mean and standard deviation.

Example: Standard normal has mean 0 and standard deviation 1.

Formula sheet

11 formulas

Factorial

n! = n (n - 1) \dots 2 \cdot 1

Variables: $n$ : nonnegative integer; $0! = 1$
Remember: Counts arrangements of $n$ distinct objects.

Permutation

_{n} P_{r} = \frac{n !}{( n - r )!}

Variables: $0 \leq r \leq n$
Remember: Use when order matters.

Combination

_{n} C_{r} = \frac{n !}{r ! ( n - r )!}

Variables: $0 \leq r \leq n$
Remember: Use when order does not matter.

Conditional probability

P (A ∣ B) = \frac{P ( A \cap B )}{P ( B )}

Variables: $P (B) > 0$
Remember: Restrict the sample space to $B$ .

Binomial probability

P (X = x) =_{n} C_{x} p^{x} (1 - p)^{n - x}

Variables: $n$ trials; constant $p$ ; $x$ successes
Remember: Independent trials with two outcomes.

Geometric probability

P (X = x) = (1 - p)^{x - 1} p

Variables: $x$ : trial of first success
Remember: Counts trials until first success.

Hypergeometric probability

P (X = x) = \frac{_{K} C _{x} _{N - K} C _{n - x}}{_{N} C _{n}}

Variables: sample $n$ without replacement from $N$ ; $K$ successes
Remember: Trials are dependent because there is no replacement.

Expected value

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

Variables: $x$ : possible value
Remember: Long-run average, not necessarily an attainable outcome.

Variance

Var (X) = \sum (x - μ)^{2} P (X = x)

Variables: $μ = E (X)$
Remember: Standard deviation is the square root of variance.

Z-score

z = \frac{x - μ}{σ}

Variables: $σ > 0$
Remember: Standardizes a value relative to its distribution.

Normal approximation correction

P (X \leq x) \approx P (Y < x + 0.5)

Variables: $Y$ : normal approximation
Remember: Use a continuity correction when approximating discrete counts.

Worked examples

Choose a committee

How many $3$ -person committees can be chosen from $8$ people?

Show solution

Order does not matter, so use combinations.
Compute $_{8} C_{3} = \frac{8 !}{3 ! 5 !}$ .
Cancel: $\frac{8 \cdot 7 \cdot 6}{3 \cdot 2 \cdot 1} = 56$ .
There are $56$ committees.

Comparison table

Common probability distributions
Distribution	Situation	Key condition
Binomial	Number of successes in $n$ trials	Independent, two outcomes, constant $p$
Geometric	Trial number of first success	Independent, two outcomes, constant $p$
Hypergeometric	Successes in a sample	Sampling without replacement
Normal	Continuous bell-shaped measurement	Model determined by $μ$ and $σ$

Variable glossary

$n$: trial or sample count
$p$: probability of success
$μ$: population mean or expected value
$σ$: population standard deviation
$r$: correlation coefficient when used in regression

Student tools

Open graph studio Guided calculators