chapter 1

Chapter 4: Continuous Random Variables and Probability Distributions∫22b. P(X > 0) = .09375(4 − x ) dx = .09375(4x− ) = . 501∫ − 12c. P(-1 < X < 1) = .09375(4 − x ) dx = . 6875d. P(x < -.5 OR x > .5) = 1 – P(-.5 ≤ X ≤ .5) = 1 -∫ . 09375(4 − x2 ) dx−= 1 - .3672 = .6328.5.53x3⎤⎥⎦204.∞∞ x 2 22 2 ∞−x/ 2θ− x / 2θa.∫ f ( x;θ ) dx == − ]0= 0 − ( −1)= 1− ∞ ∫ e dx e0 2θ200−= 2002 2x / 2θe−∞0 2b. P(X ≤ 200) =∫ f x;θ ) dx ∫x( dxθ2 2 200−x/ 2θ= − e ]0≈ −.1353+ 1 = . 8647P(X < 200) = P(X ≤ 200) ≈ .8647, since x is continuous.P(X ≥ 200) = 1 - P(X ≤ 200) ≈ .1353∫ 200100c. P(100 ≤ X ≤ 200) = ( x;) dx =200/ 20,000f θ − e ] −x 2≈ . 4712100d. For x > 0, P(X ≤ x) =∫ ∞ −xx yf ( y;θ ) dy = ∫02ee2 22 2 x2 2−y/ 2θ −y/ 2θ−x/ 2θdx = − e ]0= 1 − e5.∫ ∞ −∞f223 2x83( x)dx = ∫ kx dx = k = k ⇒ k =0a. 1 = ( )] ( )3 0 38∫1b. P(0 ≤ X ≤ 1) = ] 3 2 1 3 1x dx = x = = . 12508∫1.5810c. P(1 ≤ X ≤ 1.5) = ] 3 2 1 3 1 3 1 3 19x dx = x = ( ) − ( ) = ≈ . 2969181.521.51 3d. P(X ≥ 1.5) = 1 -∫ x dx = x ] = −1 ( 3 )88 08 2081.5188238164337[ − 0] = 1−27 = . 57813 1 ≈64 64130