Module 3 Homework Solutions - Iowa State University STAT 330

Stat 330 Online: Module 3 Homework Show all of your work, and upload this homework to Canvas. 1. Suppose a continuous random variable X has the following probability density function fX(x) =  cx 0 ≤ x ≤ 2 0 otherwise (a) Find the value of c that makes fX(x) a valid probability density function. (Recall a property that a PDF must have) (b) Give the CDF, FX(x). (c) Find P(0.5 ≤ X ≤ 1.5) using fX(x). (d) Find P(1 ≤ X ≤ 2) using FX(x). (e) Find the value of x such that the probability of being less than x is .75 (f) Find E(X). (g) Find V ar(X). Answer: (a) For a PDF to be valid, R ∞ −∞ fX(x)dx = 1. We have: Z 2 0 cx = 1 cx2 2     2 0 = 2c → 2c = 1 → c = 1 2 So the final valid PDF is fX(x) =  x 2 0 ≤ x ≤ 2 0 otherwise (b) X < 0 → FX(x) = 0, x > 2 → FX(x) = 1. For x ∈ [0, 2] we have: FX(t) = P(X ≤ t) = Z t 0 x 2 dx = x 2 4     t 0 = t 2 4 FX(t) =    0 t < 0 t 2 4 0 ≤ t ≤ 2 1 t > 2 (c) R 1.5 .5 x 2 dx = x 2 4     1.5 .5 = 1.5 2 4 − .5 2 4 = .5625 − .0625 = 0.50 (d) P(1 ≤ X ≤ 2) = FX(2) − FX(1) = 2 2 4 − 1 2 4 = 1 − .25 = 0.75 (e) We want x such that P(X ≤ x) = .7