Get Exponential Distribution - Stat Psu

The probability distribution for a random variable describes how the probabilities are distributed over the values of the random variable. For a discrete random variable, x, the probability distribution is defined by a probability mass function, denoted by f(x).

If (the Greek letter "lambda") equals the mean number of events in an interval, and (the Greek letter "theta") equals the mean waiting time until the first customer arrives, then: = 1 and. For example, suppose the mean number of customers to arrive at a bank in a 1-hour interval is 10.

Probability density function 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.

As a simple example of a probability distribution, let us look at the number observed when rolling two standard six-sided dice. Each die has a 1/6 probability of rolling any single number, one through six, but the sum of two dice will form the probability distribution depicted in the image below.

Other examples include the length, in minutes, of long distance business telephone calls, and the amount of time, in months, a car battery lasts. It can be shown, too, that the value of the change that you have in your pocket or purse approximately follows an exponential distribution.

Probability mass function. The horizontal axis is the index k, the number of occurrences. is the expected rate of occurrences. The vertical axis is the probability of k occurrences given .

is the shape parameter which indicates the average number of events in the given time interval. The following is the plot of the Poisson probability density function for four values of . Cumulative Distribution Function.

In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.

Now I define to be the rate of arrival of a bus per minute. So, = (1/10).