• PMF / PDF: Gives the probability or likelihood of an outcome.

• Cumlative Density Function (CDF): Gives the probability that a variable is less than or equal to a value.

• Quantile Function (QF): Tells you the value corresponding to a given cumulative probability.

• The Probability Mass Function (PMF) is used for discrete distributions.

• It gives the probability of a specific outcome occurring in a random trial.

For a discrete random variable \(X\), the PMF \(f(x)\) is:

\[ f(x) = P(X = x) \]

• The Probability Density Function (PDF) is used for continuous distributions.

• It gives the relative likelihood of a random variable taking on a specific value, but the probability of any single point is 0. Instead, we consider the area under the curve.

For a continuous random variable \(X\), the PDF \(f(x)\) satisfies:

\[ P(a \leq X \leq b) = \int_a^b f(x) \, dx \]

• The Cumulative Distribution Function (CDF) is the probability that a random variable \(X\) takes a value less than or equal to \(x\).

• For discrete variables:

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

• For continuous variables:

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

The CDF is always non-decreasing and ranges from 0 to 1.

• The Quantile Function (QF) is the inverse of the CDF.

• It tells you the value \(x\) such that the probability of the random variable being less than or equal to \(x\) is a given probability \(p\).

For a given \(p\), the quantile function is:

\[ Q(p) = F^{-1}(p) \]

For discrete distributions, it gives the smallest value \(x\) such that \(F(x) \geq p\).

For continuous distributions, it provides the exact value corresponding to the probability \(p\).

The Bernoulli distribution is a discrete distribution with two possible outcomes, usually labelled \(S\) for “success” and \(F\) for “failure”, it applies equally well to any two dichotomous outcomes. It is a special case of the binomial distribution where a single trial is conducted (\(n=1\)).

The probability mass function is given by:

\[ f(x|p) = p^x (1-p)^{1-x} \]

where \(x \in \{0, 1\}\) and \(p\) is the probability of success.

The Binomial distribution is a discrete distribution with \(n\) independent trials, each with two possible outcomes, usually labelled \(S\) for “success” and \(F\) for “failure”. It is defined by two parameters: the number of trials \(n\) and the probability of success \(p\).

The probability mass function is given by:

\[ f(x|n, p) = \binom{n}{x} p^x (1-p)^{n-x} \]

where \(x \in \{0, 1, \ldots, n\}\) and \(p\) is the probability of success.

• dbinom(x, size, prob) gives the probability mass function for the binomial distribution with parameters size and prob at the values x.

• pbinom(x, size, prob) gives the cumulative distribution function for the binomial distribution with parameters size and prob at the values x.

• qbinom(p, size, prob) gives the quantile function for probability p of the binomial distribution with parameters size and prob.

• 1 - pbinom(3, size=n, prob=p, lower.tail=TRUE)

• 1 - sum(dbinom(0:3, size=n, prob=p))

The normal distribution is a continuous distribution with a bell-shaped probability density function. It is defined by two parameters: the mean \(\mu\) and the standard deviation \(\sigma\).

The probability density function is given by:

\[ f(x|\mu, \sigma) = \frac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right) \]

where \(x \in \mathbb{R}\), \(\mu \in \mathbb{R}\) and \(\sigma > 0\).

• dnorm(x, mean, sd) gives the probability density function for the normal distribution with parameters mean and sd at the values x.

• pnorm(x, mean, sd) gives the cumulative distribution function for the normal distribution with parameters mean and sd at the values x.

• qnorm(p, mean, sd) gives the quantile function for probability p of the normal distribution with parameters mean and sd.

• 1 - pnorm(1, mean=mu, sd=sigma, lower.tail=TRUE)