PDF of Exponential Variate with a Specific Variance

In the realm of probability and statistics, the exponential distribution plays a significant role in modeling events that occur continuously over a period. This article delves into the specific case of an exponential distribution with a variance of 4, exploring its probability density function (PDF) and understanding the properties that define it. Let's unravel the details.

Understanding the Exponential Distribution

The exponential distribution is a continuous probability distribution that is often used to model the time between events in a Poisson process. It is characterized by a single parameter, usually denoted by lambda;, which represents the rate parameter. The PDF of the exponential distribution is given by:

[ f(x; lambda) lambda e^{-lambda x} ]

for (x geq 0) and (lambda > 0).

Variance and the Parameter lambda;

The variance of the exponential distribution is given by:

[ text{Var}(X) frac{1}{lambda^2} ]

Given that the variance of the distribution in question is 4, we can solve for the parameter lambda;.

Step-by-Step Solution

1. Start with the formula for the variance of the exponential distribution:

[ text{Var}(X) frac{1}{lambda^2} ]

2. Substitute the given variance value, which is 4:

[ 4 frac{1}{lambda^2} ]

3. Solve for (lambda).

4. Taking the square root of both sides:

[ lambda^2 frac{1}{4} ]

[ lambda frac{1}{2} ]

Deriving the PDF

Now that we know the value of (lambda), we can substitute it back into the PDF formula:

[ f(x; lambda) frac{1}{2} e^{-frac{1}{2} x} ]

Therefore, the probability density function (PDF) of the exponential variate whose variance is 4 is:

[ f(x) frac{1}{2} e^{-frac{1}{2} x} ]

Graphical Representation

Here’s a graph illustrating the PDF of the exponential variate with a variance of 4:

The graph shows the decreasing nature of the PDF as x increases, highlighting how the exponential distribution describes the time between events in a memoryless process.

Applications and Relevance

The exponential distribution with a variance of 4 has numerous applications in fields such as reliability engineering, queuing theory, and survival analysis. For instance, in reliability engineering, components that have a lifetime described by an exponential distribution with a specific variance might indicate a certain mean time to failure.

Related Questions and Further Reading

If you have any more questions about the exponential distribution or need further clarification, feel free to explore the following related topics:

Exponential Distribution Explained Understanding the Variance in Statistics Probability Density Functions for Common Distributions

Conclusion

In summary, by understanding the relationship between the variance and the rate parameter (lambda) of the exponential distribution, we were able to derive the specific PDF for an exponential variate with a variance of 4. Such distributions play a crucial role in various fields and offer valuable insights into the behavior of stochastic processes.