Integrals and Probability Density Functions

Area Under a Curve

The area under a curve between $x=a$ and $x=b$ (for a positive function) is called the integral of the function.

Figure: Example 1, 2 and 3

Details

Definition

The area under a curve between $x=a$ and $x=b$ (for a positive function) is called the integral of the function and is denoted: $\int_{a}^{b} f(x)dx$ when it exists.

The Antiderivative

Given a function $f$, if there is another function $F$ such that $F'=f$, we say that $F$ is the antiderivative of $f$. For a function $f$ the antiderivative is denoted by $\int f dx$.

Note that if $F$ is one antiderivative of $f$ and $C$ is a constant, then $G=F+C$ is also an antiderivative. It is therefore customary to always include the constant, e.g. $\int x dx=\displaystyle\frac{1}{2}x^2+C$.

Examples

Example

The antiderivative of $x$ to a power raises the power by one and divides it by the new power:

$\int x^n dx=\displaystyle\frac{1}{n+1}x^{n+1} +C$

Example

$\int e^x dx=e^x+C$

Example

$\int \displaystyle\frac{1}{x} dx=\ln(x)+C$

Example

$\int 2xe^{x^2} dx=e^{x^2}+C$

The Fundamental Theorem of Calculus

If $f$ is a continuous function, and $F'(x)=f(x)$ for $x\in[a,b]$, then $\int_a^b f(x)dx=F(b)-F(a)$

Detail

It is not too hard to see that the area under the graph of a positive function $f$ on the interval $[a,b]$ must be equal to the difference of the values of its antiderivative at $a$ and $b$. This also holds for functions which take on negative values and is formally stated below.

Definition

Fundamental theorem of calculus: If $F$ is the antiderivative of the continuous function $f$, i.e. $F'=f$ for $x\in[a,b]$, then $\int_a^b f(x)dx=F(b)-F(a)$.

This difference is often written as $\int_a^b f dx$ or $[F(x)]_a ^b$.

Examples

Example

The area under the graph of $x^n$ between $0$ and $3$ is $\int_0^3 x^n dx = [\displaystyle\frac{1}{n+1}x^{n+1}]_0 ^3=\displaystyle\frac{1}{n+1}3^{n+1}-\displaystyle\frac{1}{n+1}0^{n+1}=\displaystyle\frac{3^{n+1}}{n+1}$

Example

The area under the graph of $e^x$ between $3$ and $4$ is $\int_3^4 e^x dx =[e^x]_3 ^4= e^4-e^3$

Example

The area under the graph of $\displaystyle\frac{1}{x}$ between $1$ and $a$ is $\int_1^a \displaystyle\frac{1}{x} dx =[\ln(x)]_1 ^a= \ln(a)-\ln(1)=\ln(a)$

Density Functions

The probability density function (p.d.f.) and the cumulative distribution function (c.d.f.).

Details

Definition

If $X$ is a random variable such that

$P(a\leq X\leq b)=\int\limits^{b}_{a}f(x)dx$

for some function $f$ which satisfies $f(x)\geq0$ for all $x$ and

$\int\limits^\infty_{-\infty} f(x)dx = 1$

then $f$ is said to be a probability density function (p.d.f.) for $X$.

Definition

The function

$F(x)= \int\limits^{x}_{-\infty} f(t)dt$

is the cumulative distribution function (c.d.f.).

Examples

Example

Consider a random variable $X$ from the uniform distribution, denoted by $X\sim Unf(0,1)$. This distribution has density:

$$f(x) = \begin{cases} 1 &\text{if } 0 \leq x \leq 1 \\ 0 &\text{e.w.} \end{cases}$$

The cumulative distribution function is given by:

$$P[X\leq x] = \int\limits^{x}_{-\infty} f(t)dt = \begin{cases} 0 & \text{if } x < 0 \\ x & \text{if } 0 \leq x \leq 1 \\ 1 & \text{else} \end{cases}$$

Example

Suppose $X \sim P(\lambda)$, where $X$ may denote the number of events per unit time. The p.m.f. of $X$ is described by

$p(x)=P[X=x]=e^{-\lambda}\displaystyle\frac{\lambda^x}{x!} \text{ for } x = 0,1,2,\dots$

Let $T$ denote the waiting time between events, or simply until the first event. Consider the event $T>t$ for some number $t>0$. If $X\sim p(\lambda)$ denotes the number of events per unit time, then let $X_t$ be the number of events during the time period for $0$ through $t$. Then it is natural to assume $X_t \sim P(\lambda t)$ and it follows that $T>t$ if and only if $X_t=0$ and we obtain $P[T>t]=P[X_t=0]=e^{-\lambda t}$. It follows that the c.d.f. of $T$ is

$F_T(t)=P[T\leq t]=1-P[T>t]=1-e^{-\lambda t} \text{ for } t > 0$

The p.d.f. of $T$ is therefore

$f_T(t)=F_T'(t)=\displaystyle\frac{d}{dt}F_T(t)=\displaystyle\frac{d}{dt}(1-e^{-\lambda t})=0-e^{- \lambda t} \cdot (-\lambda)=\lambda e^{-\lambda t}$

for $t \geq 0$ and $f_T(t)=0 \text{ for } t < 0$

The resulting density

$$f(t) = \begin{cases} \lambda e^{-\lambda t} & \text{for } t \geq0 \\ 0 & \text{for } t<0 \end{cases}$$

describes the exponential distribution.

This distribution has the expected value

$E[T]=\int_{-\infty}^{\infty} tf(t)dt=\lambda \int_{0}^{\infty} t e^{-\lambda t}dt$

Let's use integration by parts (see below), i.e.: $\int fg' = fg - \int f'g$ to solve that integral. Let $f=t$ and $g'=e^{-\lambda t}$. Then $f' = 1$ and $g=-\displaystyle\frac{e^{- \lambda t}}{\lambda}$. We obtain:

$$\begin{aligned} &= \lambda \left( \left[ \displaystyle\frac{-te^{-\lambda t}}{\lambda}\right]_{0}^{\infty} - \int_0^\infty - \displaystyle\frac{-e^{-\lambda t}}{\lambda} dt \right) \\ &= \lambda \left( (0 - 0) - \left[ \displaystyle\frac{e^{-\lambda t}}{\lambda^2} \right]_0^\infty \right) \\ &= -\lambda \left(0 - \displaystyle\frac{1}{\lambda^2}\right) \\ &= \displaystyle\frac{1}{\lambda} \end{aligned}$$

Probabilities In R: The Normal Distribution

R has functions to compute values of probability density functions (p.d.f.) and cumulative distribution functions (c.m.d.) for most common distributions.

Details

The p.d.f. for the normal distribution is

$p(t)=\displaystyle\frac{1}{\sqrt{2\pi}}e^{-\displaystyle\frac{t^2}{2}}$

The c.d.f. for the normal distribution is

$\Phi(x)=\int_{-\infty}^x\displaystyle\frac{1}{\sqrt{2\pi}}e^{-\displaystyle\frac{t^2}{2}}dt$

Examples

Example

dnorm() gives the value of the normal p.d.f.

Example

pnorm() gives the value of the normal c.d.f.

Some Rules of Integration

Examples

Example

Using integration by parts we obtain:

$\int \ln(x)x dx= \displaystyle\frac{1}{2}x^2\ln(x)-\int \displaystyle\frac{1}{2}x^2\cdot \displaystyle\frac{1}{x} dx = \displaystyle\frac{1}{2}x^2\ln(x)-\int \displaystyle\frac{1}{2}x dx=\displaystyle\frac{1}{2}x^2\ln(x)-\displaystyle\frac{1}{4}x^2$

Example

Consider $\int_1^2 2xe^{x^2} dx$. By setting $x=g(t)=\sqrt{t}$ we obtain

$\int_1^2 2xe^{x^2} dx = \int_1^4 2\sqrt{t}e^{t}\displaystyle\frac{1}{2\sqrt{t}}dt=\int_1^4 e^t dt=e^4-e$

Handout

The two most common "tricks" applied in integration are a) integration by parts and b) integration by substitution

a) Integration by parts

$(fg)'=f'g+fg'$

by integrating both sides of the equation we obtain:

$fg=\int f'g dx + \int fg' dx \leftrightarrow \int fg' dx=fg-\int f'g dx$

b) Integration by substitution

Consider the definite integral $\int_a^b f(x) dx$ and let $g$ be a one-to-one differential function for the interval $(c,d)$ to $(a,b)$. Then

$\int_a^b f(x) dx=\int_c^d f(g(y))g'(y) dy$