Exponential Distribution Cumulative Probability

Using the PDF of an exponential distribution $f(x) = \lambda e^{-\lambda x}$, calculate the probability that $X$ is less than a given value $x$.

When dealing with continuous probability distributions like the exponential distribution, one of the key concepts is understanding the cumulative distribution function (CDF). The CDF is a function that gives the probability that a random variable is less than or equal to a certain value. For an exponential distribution characterized by the rate parameter lambda, the CDF can be derived by integrating the probability density function (PDF). This involves integrating the expression for the PDF, lambda times e to the power of minus lambda times x, from zero to the given value x. This integration provides a function in terms of x, which gives the cumulative probability for any value of the random variable X up to that point.

In practical terms, the result of this integration helps you understand how likely it is for the random process described by this exponential distribution to produce an outcome less than any given x. This is particularly useful in contexts where the exponential distribution is applicable, such as modeling time between events in a Poisson process. By evaluating the CDF at different values, you gain a better understanding of the distribution and behavior of your random variable, which can aid in making predictions and decisions based on probabilistic models.