Question

If X has an exponential distribution with parameter , derive a general expression for the (100p)th percentile of the distribution. Then specialize to obtain the median.Let denote the x value of the percentile. Substitute the value into the cumulative distribution function for the general exponential distribution with parameter , set the function equal to p and solve for .

Answer 1

To find the (100p)th percentile of an exponential distribution with decay parameter m, solve the equation 1 - e^-mx = p for x, resulting in x = -ln(1 - p) / m. The median is calculated when p = 0.5 and is ln(2) / m.

Step-by-step explanation:

Deriving the 100p-th Percentile of the Exponential Distribution

To find the (100p)th percentile of an exponential distribution, we can set the cumulative distribution function (CDF) equal to p, where p represents the probability.
Since the CDF of an exponential distribution with decay parameter m is P(X ≤ x) = 1 - e-mx, to find the x value for the percentile, we need to solve the following equation:

1 - e-mx = p

Solving for x, we get:

e-mx = 1 - p

-mx = ln(1 - p)

x = -× ln(1 - p) / m

This x represents the (100p)th percentile. To find the median, which is the 50th percentile (p = 0.5), we substitute p with 0.5:

x = -(1× ln(1 - 0.5) / m)

x = -(ln(0.5) / m)

x = -(ln(0.5) / m) = ln(2) / m

Thus, the median of an exponential distribution is ln(2) / m.