In each of Exercises 23-28, a function g and an interval I...

In each of Exercises $23-28,$ a function $g$ and an interval I are specified. The function $g$ is nonnegative on $I .$ Find a number $c$ such that $f(x)=\operatorname{cg}(x)$ is a probability density function on $I .$
$$ g(x)=e^{-2 x} \quad I=[0, \infty] $$

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Key Concepts

Probability Density Function

A probability density function (pdf) is a function that describes the likelihood of a continuous random variable. For a function to qualify as a pdf, its integral over its entire support (the interval where the function is defined) must equal one. This property ensures that the total probability of all outcomes is 1.

Normalization Constant

A normalization constant is a scalar value that is used to adjust a function so that it becomes a valid probability density function. By multiplying a given nonnegative function by this constant, the total area under the curve (its integral over its support) is forced to equal one, thereby satisfying the probabilistic requirement.

Integration of Exponential Functions

Integrating exponential functions, particularly those of the form e^(??x) over a semi-infinite interval, often results in a simple form involving the parameter ?. This process is critical in probability when determining the necessary scaling constant, as the integration yields a closed-form expression that can be inverted to find the normalization constant.

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