Continuous distributions are given by a **probability density function** (pdf):

\(f(x)\) is the pdf.

We require:

For a given point, this definition gives the probability being 0. This may seem counter intuitive, but assume it was non-zero. Then from the axiom of probability recall that we require a countable union of countable (disjoint) sets to have their probabilities add. Let the sets be point sets in a uniform distribution. This will result in an infinite probability.

Unlike discrete distributions, we usually don’t derive the continuous distribution from probabilistic arguments.

We sometimes treat a discrete random variable as a continuous one due to the ease in analyzing continuous functions.

## Cumulative Distribution Functions

And

To compute the probability \(P(a\le X\le b)=F(b)-F(a)\). This is a useful formula. Also, it is irrelevant whether the endpoints are included.

Let \(0\le p\le1\). The \((100p)\) th percentile is obtained by solving for \(\eta(p)\) in

The median is given by \(0.5=F(\tilde{\mu})\)

Tip: It is often easier to calculate the cdf and differentiate.

## Expected Value, Variance and Mode

The mean is given by:

and the mean for a function \(h(x)\) is:

The variance is given by:

The **mode** of a continuous distribution is the value that maximizes
\(f(x)\).

## Terminology

Consider families with two parameters, \(\theta_{1},\theta_{2}\). If
changing \(\theta_{1}\) shifts the pdf, it is a **location**
parameter. If changing \(\theta_{2}\) stretches or compresses the
pdf, it is a **scale** parameter.

## Properties

Let \(Y=h(X)\). Let \(h\) be invertible such that \(x=k(y)\). Then the pdf of \(Y\) is \(g(y)=f(k(y))|k'(y)|\)

## Jensen’s Inequality

Let \(g(x)\) be convex and differentiable. Then \(g(E(X))\le E(g(X))\)

## Chebyshev’s Inequality

Chebyshev’s inequality works for continuous distributions as well.