The life span of car stereos manufactured by a certain company is measured by a random variable X that is exponentially distributed with a probability density function $$f ( x ) = \left\{ \begin{array} { l l }
0.2 e ^ { - 0.2 x } & \text { if } x \geq 0 \
0 & \text { if } x

Question

The life span of car stereos manufactured by a certain company is measured by a random variable X that is exponentially distributed with a probability density function $$f ( x ) = \left\{ \begin{array} { l l } 
0.2 e ^ { - 0.2 x } & \text { if } x \geq 0 \
0 & \text { if } x < 0
\end{array} \right.$$ where x is the life span in years of a randomly selected stereo. What is the probability that the life span of a randomly selected stereo is between 5 and 16 years? Round to the nearest hundredth.
A) 0.33
B) 0.36
C) 0.29
D) 0.16

Quizplus · Accepted Answer

The probability that the life span is between 5 and 16 years is calculated using the cumulative distribution function (CDF) of the exponential distribution. The CDF is $ F(x) = 1 - e^{-0.2x} $. Thus, the probability is $ F(16) - F(5) = (1 - e^{-3.2}) - (1 - e^{-1}) \approx 0.333 $, which rounds to 0.33.

The Life Span of Car Stereos Manufactured by a Certain f(x)={0.2e−0.2x if x≥00 if x<0f ( x ) = \left\{ \begin{array} { l l } 0.2 e ^ { - 0.2 x } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.f(x)={0.2e−0.2x0​ if x≥0 if x<0​

The Life Span of Car Stereos Manufactured by a Certain $f ( x ) = \left\{ \begin{array} { l l } 0.2 e ^ { - 0.2 x } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$