#11 and #14 on page 303 of probability text

  1. A company buys 100 lightbulbs, each of which has an exponential lifetime of 1000 hours. What is the expected time for the first of these bulbs to burn out? \[ λ=1/1000 \] \[ n=100 \] \[ 1/(n * λ)=1/(100 * 1/1000)= 1/(1/10) = 10 hours \]

14.Assume that X1 and X2 are independent random variables, each having an exponential density with parameter λ. Show that Z = X1 − X2 has density: \[ fz(z)=(1/2)λe^(-λ|z|) \] \[ f(X_1)=λe ^(-λx_1) f(X_2)=λe ^(-λx_2) \] \[ F_Z(z) = ∫∫(f_{X_1, X_2}(x_1, x_2) dx_1 dx_2) \] (This is integrated from 0 to infinity) \[ ∫(λ^2 * e^(-λ(x_1 + x_2)) dx1 dx2) = ∫(λ e^(-λ_x1) * (-1/λ) dx_1 = \]

For x<0: \[ f_z(x)=∫e^ λ^(^x^-^2^λ^)dy=(1/2λ) e^λ^x \] x less than or equal to 0: \[ f_z(x)=∫e^ λ^(^x^-^2^λ^)dy=(1/2λ) e^λ^x(e^-^2^λ^x)=(1/2)(e^-^λ^x) \]

#1 on page 320-321

  1. Let X be a continuous random variable with mean μ = 10 and variance σ^2= 100/3. Using Chebyshev’s Inequality, find an upper bound for the following probabilities: $$
  1. P(|X-10|≥2). \[ \]
  2. P(|X-10|≥5). $$

\[ (c) P(|X-10|≥9). \] \[ (d) P(|X-10| ≥ 20). \]

Chebyshev’s inequality: \[ P(|X−μ|≥kσ)≤1/k^2 \] a. \[ σ^2= 100/3 σ=10/√3 \] \[ k=2/σ=2/(10/√3) 1/k^2 = 1/(2/(10/√3))^2 = 1/(.12) = 8.33 \] b. \[ k=5/σ=5/(10/√3) 1/k^2 = 1/(5/(10/√3))^2 = 1/(3/4) = 1.33 \] c. \[ k=9/σ=9/(10/√3) 1/k^2 = 1/(9/(10/√3))^2 = 4.12 \] d. \[ k=20/σ=20/(10/√3) 1/k^2 = 1/(20/(10/√3))^2 = 1/12 = 0.083 \]