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 1 and 11 years? Round to the nearest hundredth.
A)0.64
B)0.71
C)0.35
D)0.78

Quizplus · Accepted Answer

The probability that the life span of a randomly selected stereo is between 1 and 11 years can be calculated using the cumulative distribution function (CDF) of the exponential distribution. The CDF is given by $1 - e^{-\lambda x}$, where $\lambda = 0.2$ in this case. Thus, the probability is $P(1 \leq X \leq 11) = (1 - e^{-0.2 \times 11}) - (1 - e^{-0.2 \times 1}) = e^{-0.2} - e^{-2.2}$. Calculating this gives approximately 0.713, or 0.71 when rounded to the nearest hundredth.

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 $f ( x ) = \left\{ \begin{array} { l l } 0.2 e ^ { - 0.2 x } & \text { if } x \geq 0 \\0 & \text { if } x < 0\end{array} \right.$