$f ( x ) = \left\{ \begin{array} { l l }
7 e ^ { - 7 x } & \text { for } x \geq 0 \
0 & \text { for } x

Question

$f ( x ) = \left\{ \begin{array} { l l } 
7 e ^ { - 7 x } & \text { for } x \geq 0 \
0 & \text { for } x < 0
\end{array} \right.$ is a probability density function.

Quizplus · Accepted Answer

For a function to be a probability density function (PDF), it must satisfy two conditions: (1) the function must be non-negative for all values of $x$, and (2) the integral of the function over its entire range must equal 1. The given function $f(x) = 7e^{-7x}$ for $x \geq 0$ and $f(x) = 0$ for $x < 0$ meets both conditions. It is non-negative for all $x$, and integrating $f(x)$ from $0$ to $\infty$ yields $1$, satisfying the requirements of a PDF.

f(x)={7e−7x for x≥00 for x<0f ( x ) = \left\{ \begin{array} { l l } 7 e ^ { - 7 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.f(x)={7e−7x0​ for x≥0 for x<0​

$f ( x ) = \left\{ \begin{array} { l l } 7 e ^ { - 7 x } & \text { for } x \geq 0 \\0 & \text { for } x < 0\end{array} \right.$