Definition of random variable

A random variable \(X\) is a function that maps events in \(\Omega\) onto real numbers \[ X: \Omega \rightarrow \mathbb{R}\]

For example \(X(correct)=x_c=1\) and \(X(incorrect)=x_{inc}=0\).

Definition of cumulative distribution function F

\[F(x)=P(X \leq x)\]

We call the distribution of a random variable also a population.

Properties of F

\[ P(a < X \leq b) = F(b) - F(a)\]

\[ P(X > x) = 1 - F(x)\]

Definition of probability mass function f


F from f

\[F(x)=\sum_{x_i \leq x} f(X=x_i)\]

Definition of continous random variable and probability density function

\(X\) is continuous if exists \(f\) such that

\[f(x) \geq 0 \quad \forall x\] \[\int_{-\infty}^{\infty}f(x) dx = 1\] \[P(a \leq x \leq b) = \int_{a}^{b} f(x) dx\]

\(f\) is called the probability density function.

We can associate a pdf to discrete random variables using the delta of Dirac \[f(x)=\sum_{i=1}^{n}\delta (x-x_i) p_i\]

F from f

Given the definitions of \(F\) and \(f\) for continous random variables is follows that

\[F(x) = \int_{-\infty}^{x} f(x) dx\]

and by the fundamental theorem of calculus

\[ F'(x) = f(x)\]

Definition of inverse cumulative distribution function or quantile function

\(F^{-1}(q)\) is the unique real \(x\) such that \(F(x)=q\).

Examples of discrete random variables

The following examples correspond to parameterized families of random variables. Once we fix the parameters, we have a specific random variable of the family.


\(f_k = p^k(1-p)^{1-k}\)


\(f_k = \binom{n}{k}p^k(1-p)^{n-k}\)


If \(X_1\) is distributed binomially \((X_1 \sim Binomial(n_1,p))\) and \(X_2 \sim Binomial(n_2,p)\), then \(X_1 + X_2 \sim Binomial(n_1 + n_2, p)\)


\(f_k = \frac{\lambda^k}{k!}e^{-\lambda}\)

Examples of continous random variables

Normal (\(X \sim N(\mu,\sigma)\))


When \(\mu = 0\) and \(\sigma = 1\) we say that \(X\) has a standard normal distribution. The random variable is usually called \(Z\), the probability distribution function \(\varphi\) and the cumulative distribution function \(\Phi\).


If \(X \sim N(\mu, \sigma^2)\), then \(Z=\frac{X-\mu}{\sigma} \sim N(0,1)\).

If \(Z \sim N(0, 1)\), then \(X=\mu + Z \sigma \sim N(\mu, \sigma^2)\).

If \(X_i \sim N(\mu_i, \sigma_i^2)\) then \(\sum_{i} X_i \sim N (\sum_{i} \mu_i, \sum_{i} \sigma_i^2)\)

\(P(a < X < b) = P(\frac{a - \mu}{\sigma} < Z < \frac{b - \mu}{\sigma}) = \Phi( \frac{b - \mu}{\sigma}) - \Phi(\frac{a - \mu}{\sigma})\)

Example 1

Find \(P(X>2)\) if \(X \sim N(4,5)\).

\(P(X>2) = 1 - F(2) = 1 - \Phi(\frac{2 - 4}{\sqrt{5}})\)

1 - pnorm((2-4)/sqrt(5))
## [1] 0.8144533

Given that pnorm also accepts \(\mu\) and \(\sigma\) we can do it more directly

1 - pnorm(2, 4, sqrt(5))
## [1] 0.8144533


pnorm(2, 4, sqrt(5), lower.tail = FALSE)
## [1] 0.8144533

Example 2

Find \(F^{-1}(0.3)\) if \(X \sim N(2,6)\).


\(F^{-1}(0.3) = X = \mu + Z \sigma\)

2 + qnorm(.3) * sqrt(6)
## [1] 0.7154863

Given that qnorm also accepts \(\mu\) and \(\sigma\) we can do it more directly

qnorm(.3, 2, sqrt(6)) 
## [1] 0.7154863

Example 3

The probability from \(-2\sigma\) to \(2\sigma\) is about 0.95. For example for the standard normal distribution

1 - 2 * pnorm(-2)
## [1] 0.9544997

\(t\) distribution (\(X \sim t_\nu\))

\(f(x)=\frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})} \frac{1}{\left(1+\frac{x^2}{2}\right)^{\frac{\nu + 1}{2}}}\)

where \(\nu\) is called the degrees of freedom


  • For \(\nu \rightarrow \infty\), \(f(x)\) is normal.

  • \(\frac{\overline{X}_n - \mu}{\widehat{se}}\) is distributed \(t\) (\(\overline{X}_n\) and \(\widehat{se}\) are random variables defined later)

Cauchy distribution

\(f(x)=\frac{1}{\pi (1 + x^2)}\)


It is the \(t\) distribution for \(\nu=1\).

\(\chi^2\) distribution (\(X \sim \chi_p^2\))



If \(Z_i \sim N(0,1)\) then \(\sum_{i=1}^{p} Z_i^2 \sim \chi_p^2\).

\(F\) distribution (\(X \sim F\))

\(f(x)=\frac{\sqrt{\frac{ \left( \nu_1 x \right)^{\nu_1} \nu_2^{\nu2}}{\left( \nu_1 \, x + \nu_2 \right)^{\nu_1+\nu_2}} }}{x \, B \left( \frac{\nu_1}{2} , \frac{\nu_2}{2} \right)}\)


If \(X_1\) is distributed \(\chi^2\) with \(\nu_1\) degrees of freedom and \(X_2\) is distributed \(\chi^2\) with \(\nu_2\) degrees of freedom and they are indepedent, then \(F =\frac{X_1 / \nu_1}{X_2 / \nu_2}\) is distributed F, that is, with the \(f(x)\) above.

Bivariate distributions

Definition of joint probability mass function

If X and Y are discrete random variables

\[f(x,y) = P(X = x, Y = y)\]

Definition of joint probability density function

\[f(x,y) \geq 0 \quad \forall x\] \[\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) dx dy = 1\] \[P((X,Y) \in A) = \int \int_{A} f(x,y) dx dy\]

Definition of joint cumulative density function

\[ F(x, y) = P(X \leq x, Y \leq y) \]

Definition of marginal mass functions

If X and Y are discrete random variables

\[ f_X(x) = P(x=X)=\sum_y P(X=x, Y=y)= \sum_y f(x,y)\] \[ f_Y(y) = P(y=Y)=\sum_x P(X=x, Y=y)= \sum_x f(x,y)\]

Definition of marginal probability density functions

If X and Y are continuous random variables

\[ f_X(x) = \int_{-\infty}^{\infty} f(x,y) dy \] \[ f_Y(y) = \int_{-\infty}^{\infty} f(x,y) dx \]

Definition of independency

X and Y are independent random variables if

\[P(X \in A, Y \in B)=P(X \in A)P(Y \in B)\]

Defintion of conditional probability mass functions

\[f_{X|Y}(x|y) = P(X=x|Y=y)=\frac{P(X=x,Y=y)}{P(Y=y)}=\frac{f_{X,Y}(x,y)}{f_Y(y)}\]

Defintion of conditional probability density functions

\[ f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)}\]

Multivariate distributions and iid samples

Definition of random vector

\[ X=(X_1,X_2,\dotsc,X_n)\]

Definition of independency

\(X_1,\dotsc,X_n\) are independent if

\[P(X_1 \in A_1, \dotsc, X_n \in A_n)= \prod_{i=1}^{n} P(X_i \in A_i)\]

or equivalently

\[f(x_1,\dotsc,x_n)= \prod_{i=1}^{n} f_{X_i}(x_i)\]

Definition of iid samples

\(X_1,\dotsc,X_n\) are iid (independent and identically distributed) if

  • \(X_1,\dotsc,X_n\) are independent

  • \(X_1,\dotsc,X_n\) have the same marginal cumulative distribution function \(F\) (or \(f\))

We say that \(X_1,\dotsc,X_n \sim F\)

and call \(X_1,\dotsc,X_n\) a random sample of size n from \(F\). Much of statistics deals with random samples.



Multivariate normal


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Knoblauch, K., & Maloney, L. T. (2012). Modeling Psychophysical Data in R. New York: Springer.

Wasserman, L. All of statistics (2004). Springer Science & Business Media.