chapter 1

Chapter 4: Continuous Random Variables and Probability Distributions108.1−α∞ k 51= ∫ dx = k ⋅ ⇒ k = α −1 55 αx α −11−αa. ( )where we must have α > 1.b. For x ≥ 5, F(x) =x∫5ykαdy = 51−α⎡ 1⎢⎣51−α−x1α−1⎤ ⎛ 5 ⎞⎥ = 1 − ⎜ ⎟⎦ ⎝ x ⎠α−1.c. E(X) =∫∞k∞ kkx ⋅ dx = x ⋅ dx =, provided α > 2.∫( α 2)5 α 5 α−1α−xx 5 2 ⋅ −⎛ ⎛ X ⎞ ⎞ ⎛ X y ⎞yyd. P ⎜ln⎜ ⎟ ≤ y⎟= P⎜≤ e ⎟ = P( X ≤ 5e) = F( 5e)⎝−⎝ 5 ⎠−( α−1)y1 e⎠⎝5⎠, the cdf of an exponential r.v. with parameter α - 1.⎛ 5= 1 − ⎜⎝ 5ey⎞⎟⎠α−1109.⎛ I o⎞a. A lognormal distribution, since ln ⎜⎟ is a normal r.v.⎝ I i ⎠P I⎛ I ⎛⎞ ⎛ ⎛ ⎞ ⎞o⎞ ⎛ I⎜o⎞I> = ⎜ > ⎟ = ⎜ ⎟ > ⎟ = − ⎜o2IP 2 P ln ln 2 1 P ln ⎜ ⎟ ≤ ln 2b. ( )⎟ o i ⎜ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ ⎜ ⎟⎝ Ii ⎠ ⎝ ⎝ Ii⎠ ⎠ ⎝ ⎝ Ii⎠ ⎠⎛ ln 2 −1⎞1 − Φ⎜⎟ = 1− Φ − =⎝ .05 ⎠⎛ I ⎞1+.0025/ 2c. E⎜⎟ = e = 2.72,⎝ Ii⎠( 6.14) 1⎛+I⎜⎝ Io o 2 .0025 .0025Var = e ⋅ ( e −1) = . 0185i⎞⎟⎠170