Exponential Distribution

Section 8.3 Exponential Distribution

Once again, consider a Poisson Process where you start with an interval of variable length X so that X measures the interval needed in order to obtain a first success with

R = (0, \infty) .

$R = (0,\infty)\text{.}$ The resulting distribution of X will be called an Exponential distribution.

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To derive the probability function for this distribution, consider finding f(x) by first considering F(x).

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Theorem 8.3.1. Exponential Probability Function.

For an exponential random variable $X\text{,}$

f (x) = λ e^{- λ x},

$\begin{equation*} f(x) = \lambda e^{-\lambda x}, \end{equation*}$

where $\lambda$ is a given parameter.

(Note that we will find later that $\lambda = \frac{1}{\mu}$ or the reciprocal of the theoretical mean.)

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Proof.

\begin{align*} F(x)& = P(X \le x)\\ & = 1 - P(X \gt x)\\ & = 1 - P(\text{first change occurs after an interval of length} x)\\ & = 1 - P(\text{no changes in the interval} [0,x])\\ & = 1 - \frac{(\lambda x)^0 e^{-\lambda x}}{0!}\\ & = 1 - e^{-\lambda x} \end{align*}

where the discrete Poisson Probability Function is used to answer the probability of exactly no changes in the "fixed" interval [0,x].

Using this distribution function and taking the derivative yields

\begin{equation*} f(x) = F'(x) = \lambda e^{-\lambda x}. \end{equation*}

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We found that if

λ

$\lambda$ is the parameter for the Poisson process, then the Poisson distribution had mean

λ T

$\lambda T$ or if one presumed that T=1, then for that distribution, the mean is just

λ .

$\lambda\text{.}$ We will find out below that the mean for the exponential distribution will be

\frac{1}{λ} .

$\frac{1}{\lambda}\text{.}$ Therefore, we will eventually present this formula using the exponental mean

μ = \frac{1}{λ}

$\mu = \frac{1}{\lambda}$ rather than using

λ

$\lambda$ in the actual formula.

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You will also often find that exercises in other textbooks and in online WeBWorK will just provide

λ

$\lambda$ for the underlying Poisson process and that the time interval T will be presumed to be T=1. For those problems, if the exercise asks for a Poisson probability, use

μ = λ

$\mu = \lambda$ while if the exercise asks for an Exponential probability, use

μ = \frac{1}{λ} .

$\mu = \frac{1}{\lambda}\text{.}$

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Theorem 8.3.2. Verification of Exponential Probability Function.

\int_{0}^{\infty} λ e^{- λ x} d x = 1

$\begin{equation*} \int_0^{\infty} \lambda e^{-\lambda x} dx = 1 \end{equation*}$

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Proof.

\begin{align*} & \int_0^{\infty} \lambda e^{-\lambda x} dx\\ & = \int_0^{\infty} e^{-u} du\\ & = -e^{-u} \big |_0^{\infty} = 1 \end{align*}

where we used the substitution $u = \lambda x\text{.}$

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Theorem 8.3.3. Derivation of Statistics for Exponential Distribution and Plotting.

μ = \frac{1}{λ}

$\begin{equation*} \mu = \frac{1}{\lambda} \end{equation*}$

σ^{2} = μ^{2}

$\begin{equation*} \sigma^2 = \mu^2 \end{equation*}$

γ_{1} = 2

$\begin{equation*} \gamma_1 = 2 \end{equation*}$

γ_{2} = 9

$\begin{equation*} \gamma_2 = 9 \end{equation*}$

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Proof.

For the mean, use integration by parts with $u = x$ and $dv = \lambda e^{-\lambda \cdot x}$ to get (eventually)

\begin{align*} \text{Mean} = \mu & = \int_0^{\infty} x \cdot \lambda e^{-\lambda \mu} dx\\ & = \left [ -(x+\frac{1}{\lambda}) e^{-\lambda \cdot x} \right ] \big |_0^{\infty} \\ & = \frac{1}{\lambda}. \end{align*}

and so the use of $\lambda = \frac{1}{\mu}$ in f(x) is warranted.

The remaining statistics are derived similarly using repeated integration by parts. The interactive Sage cell below calculates those for you automatically.

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Corollary 8.3.4. Alternate Form for the Exponential Distribution Probability Function.

Given a Poisson process and a constant $\mu = \frac{1}{\lambda}\text{,}$ suppose $X$ measures the variable interval length needed until you get a first success. Then $X$ has an exponential distribution with probability function

f (x) = \frac{1}{μ} e^{- \frac{x}{μ}} .

$\begin{equation*} f(x) = \frac{1}{\mu} e^{-\frac{x}{\mu}}. \end{equation*}$

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Proof.

Simply replace the mean as derived above in the original function.

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# Exponential Distribution
var('x,mu')
assume(mu>0)
​
f(x) =e^(-x/mu)/mu
mu = integral(x*f,x,0,oo).factor()
M2 = integral(x^2*f,x,0,oo).factor()
M3 = integral(x^3*f,x,0,oo).factor()
M4 = integral(x^4*f,x,0,oo).factor()
​
pretty_print('Mean = ',mu)
​
v = (M2-mu^2).factor()
pretty_print('Variance = ',v)
stand = sqrt(v)
​
sk = (((M3 - 3*M2*mu + 2*mu^3))/stand^3).simplify()
pretty_print('Skewness = ',sk)
​
kurt = (M4 - 4*M3*mu + 6*M2*mu^2 -3*mu^4).factor()/stand^4
pretty_print('Kurtosis = ',(kurt-3).factor(),'+3')
@interact
def _(m = slider(1,12,1/2,2,label='mu')):
    plot(f(mu=m),x,0,30).show(ymax=1)

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Theorem 8.3.5. Distribution function for Exponential Distribution.

F (x) = 1 - e^{- \frac{x}{μ}}

$\begin{equation*} F(x) = 1 - e^{-\frac{x}{\mu}} \end{equation*}$

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Proof.

Using $f(x) = \frac{1}{\mu} e^{-\frac{x}{\mu}}\text{,}$ note

\begin{align*} F(x) & = \int_0^x \frac{1}{\mu} e^{-\frac{u}{\mu}} du\\ & = - e^{-\frac{u}{\mu}} \big |_0^x\\ & = 1 - e^{-\frac{x}{\mu}} \end{align*}

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Example 8.3.6. Router Requests Revisited.

Once again, let's consider a router which, over time, has been shown to receive on average 1000 such requests in any given 10 minute period during regular working hours. This would mean that, on average, it would take $\mu = \frac{10}{1000} = \frac{1}{100} = 0.01$ minutes (i.e., less than a second) to receive the first request. If X were to measure the time interval until the first actual request comes in, then the Exponential distribution would be a good model using

f (x) = \frac{1}{0.01} e^{- \frac{x}{0.01}} .

$\begin{equation*} f(x) = \frac{1}{0.01} e^{-\frac{x}{0.01}}. \end{equation*}$

Let's determine the probability that a first request arrives in the next two seconds. First, note that since X is a continuous variable that f(x) is NOT the probability of exactly X minutes but you must integrate to compute all probabilities. Also, the next 2 seconds is actually the next $\frac{2}{60} = \frac{1}{30}$ of a minute. Therefore, F(x) is what you need in general and you find

P (X \leq \frac{1}{30}) = F (\frac{1}{30}) = 1 - e^{- \frac{\frac{1}{30}}{0.01}} \approx 0.96433.

$\begin{equation*} P(X \le \frac{1}{30}) = F(\frac{1}{30}) = 1 - e^{-\frac{\frac{1}{30}}{0.01}} \approx 0.96433. \end{equation*}$

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Checkpoint 8.3.7. WebWork - Exponential.

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r=1                # the number of successes desired
            # the mean till first must be given
mu = 3
sdev = sqrt(mu)  # the formula for the standard deviation
​
M = mu*3   # the space is infinite but we just go out 3 standard deviations
X = 0:M    # quantiles for the space R of the random variable 
​
Pexp <- function(x){dexp(x, mu )}  # create the probability function over X
​
curve(Pexp, from=0, to=M, xlab="X", col="blue", lwd=3,
 main="Gamma Sampling vs Gamma Curve vs Approximating 'Bell Curve'") 
Pnormal <- function(X){dnorm(X, mean=mu, sd=sdev)}   # to overlap a bell curve
curve(Pnormal, col="red", lwd=2, add=TRUE) 
​
Psample = rexp(10^6, mu)  # to create a histogram, sample a lot
# Xtop=max(Psample)          # for scaling the x-axis. Shift by 1/2 below.
hist(Psample, prob=TRUE, add=TRUE)

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Theorem 8.3.8. The Exponential Distribution yields a continuous memoryless model..

If X has an exponential distribution and a and b are nonnegative integers, then

$\begin{equation*} P( X > a + b \; \; | \; \; X > b ) = P( X > a) \end{equation*}$

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Proof.

Using the definition of conditional probability,

\begin{align*} P( X > a + b | X > b ) & = P( X > a + b \cap X > b ) \ P( X > b)\\ & = P( X > a + b ) / P( X > b)\\ & = e^{-(a+b)/ \mu} / e^{-b / \mu}\\ & = e^{-a/ \mu}\\ & = P(X > a) \end{align*}

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Essentials of Mathematical Probability and Statistics

Section 8.3 Exponential Distribution

Theorem 8.3.1. Exponential Probability Function.

Proof.

Theorem 8.3.2. Verification of Exponential Probability Function.

Proof.

Theorem 8.3.3. Derivation of Statistics for Exponential Distribution and Plotting.

Proof.

Corollary 8.3.4. Alternate Form for the Exponential Distribution Probability Function.

Proof.

Theorem 8.3.5. Distribution function for Exponential Distribution.

Proof.

Example 8.3.6. Router Requests Revisited.

Checkpoint 8.3.7. WebWork - Exponential.

Theorem 8.3.8. The Exponential Distribution yields a continuous memoryless model..

Proof.

Checkpoint 8.3.9.