What is EXP in statistics

Exponential distribution

Here you can find out everything about Exponential distribution including application examples, special features and the associated formulas for the density, the Distribution function and the Expected value.

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Exponential distribution statistics

The exponential distribution is a probability distribution for determining random time intervals. It is mostly used for waiting or downtime, such as the length of a phone call, the radioactive decay of atoms or the life of your cell phone.

It is the continuous version of the geometric distribution. Both are exponentially decreasing functions.

Exponential distribution formula

It is clear that the distribution is only defined for all positive real numbers, after all, negative times are not possible.

In statistics, the exponential distribution is abbreviated as follows:

You see, is our only parameter. In the following we will go into more detail on the specific formulas for calculating the probabilities.

Dense exponential distribution

The following notation of the density function looks rather confusing at first glance, but is only 100% mathematically correct in this form. This is because the exponential distribution only considers the probability of positive, real numbers.

The density function looks like this:

Density function:

Distribution function exponential distribution

As with the density function, probabilities can only be calculated with the cumulative distribution function in this context for values ​​from the positive range.

Distribution function:

Exponential distribution expectation

The expected value shows that a large lambda results in a small average time interval and the mode shows that the density function has its maximum at zero. This also makes sense, because at the beginning of the time measurement, the probability that your cell phone will still work, for example, is highest. We also list the most important dimensions for you here:

Exponential distribution example

All right, let's calculate the remaining life of the cell phone. We know from the manufacturer that the service life with

is exponentially distributed.

Calculate exponential distribution probability

here is the failure rate. So 0.1 percent of cell phones give up their ghost every day. This means that, on average, a device fails every 1000 days after commissioning.

Since you are planning a big trip this year and you couldn't afford a new cell phone, you now want to know how likely it is that your device will work for another year.

So we take the distribution function and use our values. Here's how you can calculate the probability:

We see that after 365 days, 30.6 percent of the cell phones have failed. So the probability that yours will still work is almost 70 percent. Very good! Nothing stands in the way of your journey.

Memorylessness of the exponential distribution

An important property of this probability distribution is its memorylessness. For example, in order to calculate the remaining service life of your mobile phone in the event of a memoryless distribution, you do not have to know how long the device has been in use. The probability that it will survive the next unit of time after 3 years is just as high as when it was first used. It is believed that age is irrelevant to these types of calculations. If a component with an exponentially distributed service life is not yet defective at any given point in time, the probability that it will still be functional in the next time unit is just as high as it was at the beginning. Therefore, this distribution cannot be used for the life expectancy of living beings, since age definitely plays a role here. An 80-year-old is much less likely to live another 50 years than a newborn.

Summary and most important formulas

Finally, we will show you all the important formulas again here. When you have mastered these, your statistics exam will also be like vacation.