Question

What is the probability density function (PDF) for the given cumulative distribution function (CDF) f(x) = 1 - e^(-x/50) for x > 0? Option 1: f(x) = (1/50)e^(-x/50) Option 2: f(x) = (1/50)e^(-x) Option 3: f(x) = (1/50)e^(50x) Option 4: f(x) = (1/50)e^(x)

Answer 1

Explanation:

Given CDF: F(x) = 1 - e^(-x/50) for x > 0

Now, find the PDF (f(x)):

f(x) = d/dx [F(x)]

f(x) = d/dx [1 - e^(-x/50)]

Using the chain rule:

f(x) = 0 - (-1/50) * (-e^(-x/50))

f(x) = (1/50)e^(-x/50)

So, the correct PDF is:

Option 1: f(x) = (1/50)e^(-x/50)