chapter 1

Chapter 4: Continuous Random Variables and Probability Distributions10.a.θk∞∞∞ kθ⎛ 1 ⎞⎤= ∫+⎜ − ⎟−∞∫θkk ⎥x ⎝ x ⎠⎦b. ( ; , ) == = 11 =kf x k θ dx dx θ ⋅θθθkkc. P(X ≤ b) =kθkb∫ θk+1xdx = θk⎛ 1 ⎞⎤⋅ ⎜ −k⎟⎥⎝ x ⎠⎦bθ⎛θ⎞= 1 − ⎜ ⎟⎝ b ⎠kd. P(a ≤ X ≤ b) =kθ k ⎛ 1 ⎞⎤dx = θ ⋅ ⎜ −k⎟⎥x ⎝ x ⎠⎦kb∫ ak+1ba⎛θ⎞= ⎜ ⎟⎝ a ⎠k⎛θ⎞− ⎜ ⎟⎝ b ⎠kSection 4.211.a. P(X ≤ 1) = F(1) = 1= . 254b. P(.5 ≤ X ≤ 1) = F(1) – F(.5) = 3= . 187515c. P(X > .5) = 1 – P(X ≤ .5) = 1 – F(.5) = = . 9375~ )µ ~42~2d. .5 = F ( µ = ⇒ µ = 2 ⇒ µ = 2 ≈1.414e. f(x) = F′(x) = 2 x for 0 ≤ x < 2, and = 0 otherwise∫ ∞ −∞∫21612~2f. E(X) = x ⋅ f ( x)dx = x ⋅ xdx = x dx = = ≈ 1. 333∫ ∞ −∞01612∫2 1 1= ∫02 2 ∫023x ⎤⎥6 ⎦g. E(X 2 223) = x f ( x)dx x xdx = x dx = = 2,So Var(X) = E(X 2 ) – [E(X)] 2 8 8= ( ) = ≈ . 222h. From g , E(X 2 ) = 2226 3620204x ⎤⎥8 ⎦− , σ x ≈ .4712086133