Exponential Distribution: Overview, Questions, Preparation

Probability distribution is the list of all possible outcomes of a random variable along with their corresponding probability value. Exponential distribution is used to model the probability distribution of the time periods between the process in which events occur continuously at the fixed rate.

Recognize the Exponential Distribution

In Exponential distribution, the corresponding probability value of the random variable occurs in such a way that the probability of occurrence will be larger for fewer samples and it will be smaller for larger samples. 

For example, If the amount of money customers spend in one trip to the supermarket follows an exponential distribution then there will be more people who spend small amounts of money and fewer people who spend large amounts of money. 

Derivation of PDF of Exponential Distribution

The Poisson process is a model for a series of events where the average time between events is known, but the exact timing of events is random. (You will learn more about the Poisson process in higher studies)

This is the probability of getting k events during a period of time t  with ƛ as the parameter. (ƛ is basically the rate of arrival of any event)

Considering this definition of the Poisson process, exponential distribution will be the amount of time until not a single event occurs. (During the waiting period between two events). since no events have occurred, thus we will put K=0 in equation 1, we will get:

[P(0) is the probability of zero successes in the “t” time unit.]

Equation 4 is the PDF of Exponential Distribution.

Note:- Probability Density Function is the derivative of Cumulative Distribution Function.

Exponential Distribution Graph

Three different exponential distribution are shown below:

In the graph, you can observe that the probability value of the random variable X is occurring in such a way that there are fewer large values and more small values.

What does the Memoryless property of Exponential Distribution mean?
It means, if any product has lasted for ten years, then memoryless means that the probability of its durability for the next three years (so, a total of 13 years) is going to be the same as that of a new product.

1. Let X= amount of time a shopkeeper spends with his customer follows exponential distribution with the average amount of time equal to 4 minutes. Find the probability that the shopkeeper is going to spend 5 minutes with the customer?

Solution.

 2. The amount of time a student takes to solve any problem follows an exponential distribution with the average amount of time equal to 8 minutes. What will be the probability that he will take 5 minutes to solve the problem? 

 

Solution.

A: Exponential distribution deals with the amount of time for which a product lasts and is often used to model the longevity of an electrical or mechanical device. The measurement of radioactive decay can be modelled through exponential distribution.

A: The exponential distribution is skewed to the right, with no negative values and it will contain many observations relatively close to 0 and a few observations to the right from 0.

A: In exponential distribution. Every instant is like the beginning of a new random period, which has the same distribution regardless of how much time has already elapsed. The exponential is the only memoryless continuous random variable.

A:   P(T>13|T>10)=P(T>3)

A: The sum of two Exponentially distributed random variables with same rate parameter results into a Gamma distribution. 

Q: When would you use an exponential distribution?

Q: How do you know if data is exponentially distributed?

Q: Why is exponential distribution memoryless?

Q: How do you find the CDF of an exponential distribution?

Q: What is the sum of two exponential random variables?