Stat 309 Final

The Cauchy cumulative distribution function is F(x) = (1/2) + (1/π) tan−1(x), −∞ < x < ∞

b. Find the density function.

c. Find x such that P(X > x) = .1 −∞

b. PDF: f(x) = 1/[π(1+x²)]

c. X = 3.078

Suppose that X has the density function f(x) = cx² for 0 ≤ x ≤ 1 and f(x) = 0 otherwise.

a. Find c.

b. Find the cdf.

c. What is P(.1 ≤ X <.5)?

a. c = 3

b. CDF: F(x) = { 0 : x < 0, x³ : 0 ≤ x ≤ 1, 1 : x > 1

c. P(.1 ≤ X <.5) = 0.124

Let X be a continuous random variable with the density function f(x) = 2x, 0 ≤x ≤1

a. Find E(X).

b. Find E(X²) and Var(X).

a. E(X) = 0.667

b. E(X²) = 0.5, Var(X) = 0.0556

Let X be uniformly distributed on the interval [1,2]. Find E(1/X). Is E(1/X) = 1/E(X)?

Not equal because E(1/X) = 0.693 and 1/E(X) = 0.667.

Suppose that the lifetime of an electronic component follows an exponential distribution with λ = .1.

a. Find the probability that the lifetime is less than 10.

b. Find the probability that the lifetime is between 5 and 15.

c. Find t such that the probability that the lifetime is greater than t is .01.

a. P(X<10) = 0.632

b. P(5<X<15) = 0.383

c. t = 46.052