Question

Let the random variable be the time in seconds between incoming telephone calls at a busy switchboard. Suppose that a reasonable probability model for X is given by the pdf student submitted image, transcription available below (a) Find the value of constant c so that above function properly defines a pdf. (b) What is the probability that the time between successive phone calls exceeds 4 seconds? (c) Find the cumulative distribution function of X.

Answer 1

To find the value of constant c, we need to integrate the given function over its entire domain [0, ∞) and set it equal to 1. Solving the integral equation, we find that the value of c is -1.

Step-by-step explanation:

To find the value of constant c so that the function properly defines a probability density function (pdf), we need to ensure that the integral of the pdf over its entire domain is equal to 1. In this case, the domain is [0, ∞). So, we need to integrate the given function from 0 to ∞ and set it equal to 1. After integrating, we get:

c * ∫(2e^(-2x) - e^(-4x)) dx = 1

After performing the integration, we get:

c * (-e^(-2x) + 0.5e^(-4x)) ∣ 0 to ∞ = 1

Since the limits of integration go to ∞, the second term goes to 0. Therefore, we have:

-c * e^(-2x) ∣ 0 to ∞ = 1

c * e^(-2x) ∣ 0 to ∞ = 1

Since e^(-∞) = 0, we have:

c * 0 - c * e^(-2 * 0) = 1

c * 0 - c * 1 = 1

-c = 1

c = -1

So, the value of constant c that makes the function a valid pdf is -1.