A customer service representative may spend varying amounts of time with each customer to address different issues. The time spent with each customer can be modeled by the following distribution: X ~ Exp(0.2). Find the 40th percentile.
Social Studies · College · Wed Jan 13 2021
Answered on
To find the 40th percentile of the exponential distribution X ~ Exp(λ) with rate λ = 0.2, we will use the formula for the cumulative distribution function (CDF) of the exponential distribution. The CDF is given by:
\[ F(x) = 1 - e^{-λx} \]
Here, we are trying to find the value of x such that F(x) = 0.40. We will set up the equation and solve for x:
\[ 0.40 = 1 - e^{-0.2x} \]
Solving for x, we get:
\[ e^{-0.2x} = 1 - 0.40 \] \[ e^{-0.2x} = 0.60 \]
Taking the natural logarithm of both sides, we have:
\[ -0.2x = ln(0.60) \]
Now we divide both sides by -0.2 to solve for x:
\[ x = \frac{ln(0.60)}{-0.2} \]
Using a calculator, we find the natural logarithm of 0.60:
\[ x = \frac{ln(0.60)}{-0.2} \approx \frac{-0.5108}{-0.2} \] \[ x \approx 2.554 \]
So the 40th percentile of the exponential distribution with rate 0.2 is approximately 2.554.