Presented by:Rajendr a KhemkaEnrollment ID:140420705005

IntroductionCDF (Cumulative Distribution Function), and PDF (Probability Density Function) of an exponential functions are discussed and MATLAB toolbox, user code implementation is performed.

Simulation results and Applications are discussed at end of the presentation.

Equation for PDF: ex

Equation for CDF: 1 ex

EXPONENTIAL DISTRIBUTION In probability theory and statistics, the exponential distribution (a.k.a. negative exponential distribution) is the probability distribution that describes the time between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate. It is the continuous analogue of the geometric distribution, and it has the key property of being memory less. In addition to being used for the analysis of Poisson processes, it is found in various other contexts.

PDF (Probability Density Function)The probability density function (pdf) of an exponential distribution is

Here > 0 is the parameter of the distribution, often called the rate parameter. The distribution is supported on the interval [0, ). If a random variable X has this distribution, we write X ~ Exp()The exponential distribution exhibits infinite divisibility.

CDF (Cumulative Distribution Function)

The cumulative distribution function (cdf) of an exponential distribution is

Equation for CDF: 1 ex x >= 0

For all other points the value of CDF assumes a null

MATLAB CODE% exponential distribution pdf and cdf scripting% both mean and mu should have same index clcclear all mean = input('enter the mean value for dist. function : ');mu = input('enter the mu value for dist. function : '); temp= size(mean);n= temp(1,2); % to implement PDF (Prob. Density Function)disp('Exponential PDF is : '); for i = 1:n y(i) = [2.718281828 ^ -(mean(i)./mu(i))]./ mu(i); endsubplot(1,2,1);

plot(y)title('PDF plot');axis('square'); % To implement CDF (Cumulative Dist. Function)% Function implemented is y(mu, mean) = 1 - e ^ (mean/mu)% requirement : mean and mu must be of same size disp('Exponential CDF is : ');for i = 1:n z(i) = 1 - [2.718281828 ^ -(mean(i)./mu(i))]; endsubplot(1,2,2); plot(z)title('CDF plot');axis('square');

SIMULATION RESULTS

CDF and PDF for n = 50

CDF and PDF for n = 100

CDF and PDF for n = 1000

RESULTS USING DISTTOOL - PDF

RESULTS USING DISTTOOL - CDF

APPLICATIONS

The time it takes before your next telephone call For example, the rate of incoming phone calls differs according to the time of day. But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives.

The time until a radioactive particle decays, or the time between clicks of a Geiger counter

Geiger counter is used to measure the radioactivity of a given sample and it clicks when it detects radioactivity. Higher activity corresponds to higher clicks and vice versa. Since law of radioactivity follows a decaying exponential graph, it can be taken as an example of exponential distribution

The time until default (on payment to company debt holders) in reduced form credit risk modeling, application based on financial markets.

In Queuing theory, the service times of agents in a system (e.g. how long it takes for a bank teller etc. to serve a customer) are often modeled as exponentially distributed variables.

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