Verify that the functions are probability density functions...

Verify that the functions are probability density functions for a continuous random variable $X$ over the given interval. Determine the specified probability.
$f(x)=x e^{-x}$ over $[0, \infty), P(1 \leq X \leq 3)$

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Key Concepts

Probability Calculation via Integration

Once a valid probability density function is established, the probability that the random variable falls within a particular interval is determined by integrating the pdf over that interval. This method involves computing the area under the curve of the pdf between the specified bounds, which represents the likelihood of the variable being within that range.

Integration for Verifying the Pdf

To confirm that a function qualifies as a probability density function, one must evaluate its definite integral over the entire range of possible values. The integration should yield a total area of one, which represents the total probability of 1 for the continuous random variable.

Probability Density Function

A probability density function (pdf) is a non-negative function that describes the likelihood of a continuous random variable taking on a particular value. For a function to be a valid pdf, it must integrate to one over its entire domain, ensuring that the total probability is equal to 100%.

Continuous Random Variable

A continuous random variable is a variable that can take on an infinite number of values within a given interval. Instead of assigning probabilities to individual points, probabilities are determined over intervals using areas under a probability density function.

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