chapter 1

Chapter 4: Continuous Random Variables and Probability Distributions52. P(X ≤ µ + σ[(100p)th percentile for std normal])⎛ X − µP ⎜ ≤⎝ σ⎞[...] ⎟⎠= P(Z ≤ […]) = p as desired53.⎛ ( y − b)⎞a. F y (y) = P(Y ≤ y) = P(aX + b ≤ y) = P⎜X ≤ ⎟⎝ a ⎠Now differentiate with respect to y to obtainf y (y) =F121 − [ y−(aµ+ b)]′ 2 22aσy( y)= eand variance a 2 σ 2 .2πaσ(for a > 0).so Y is normal with mean aµ + bb. Normal, mean 9 (115) + 32 = 239 , variance = 12.96554.⎛83+ 351 + 562 ⎞⎜⎟⎝ 703 + 165 ⎠a. P(Z ≥ 1) ≈ . 5 ⋅ exp= . 1587⎛ − 2362 ⎞⎜ ⎟⎝ 399.3333⎠b. P(Z > 3) ≈ . 5 ⋅ exp= . 0013⎛ − 3294 ⎞⎜ ⎟⎝ 340.75 ⎠c. P(Z > 4) ≈ . 5 ⋅ exp = . 0000317 , soP(-4 < Z < 4) ≈ 1 – 2(.0000317) = .999937⎛ − 4392 ⎞⎜ ⎟⎝ 305.6 ⎠d. P(Z > 5) ≈ . 5 ⋅ exp = . 00000029148