chapter 1

Chapter 4: Continuous Random Variables and Probability Distributions17.x=−∞x−3=2a. For 2 ≤ X ≤ 4, F( X ) ∫ f ( y)dy = ∫3 [1 −(y − 3) ] dy (let u = y-3)43x−3⎡ ⎤ ⎡3−−2 3 u 3 7 ( x 3)∫ 4[1 u ] du = ⎢u− ⎥ = ⎢x− −−14 ⎣ 3 ⎦ 4− ⎣ 3 31⎧ 0x < 2⎪⎨1 34[3x− 7 − ( x − 3) ] 2 ≤ x ≤ 4⎪⎩ 1x > 4F(x) =2x3⎤⎥⎦. Thusb. By symmetry of f(x), µ ~ = 343 12( y + 3)(1−y )−1124y 3 y ⎤2c. E(X) =∫ x ⋅ 1−( x −3)] dx =344 ∫2=34[ dx⎡⎢3y+⎣2− y−4⎥⎦−1∞42V(X) = x − µ ) f ( x)dx = ( x − 3)∫= 2=34⋅ 4 = 3322( ∫ ⋅ [1 − ( x − 3) ] dx−∞423 12 2 3 4 1(1 ).4∫ y − y dy = ⋅ = =− 14 15 518.a. F(X) =x − AB − A= p ⇒ x = (100p)th percentile = A + (B - A)p2B 1 1 x ⎤ 1 1 2 2b. E(X ) x ⋅ dx = ⋅ = ⋅ ⋅ ( B − A )= ∫⎥A B − A B − A 2 ⎦ 2 B − AA222 1 1 3 3 A + AB + BE(X ) = ⋅ ⋅ ( B − A ) =3 B − A3B=A + B22⎛ A + AB + BV ( X ) =⎜⎝ 3222( A B) ⎞ ( B − A) = ,⎞ ⎛ +⎟ − ⎜⎠ ⎝ 2⎟⎠12σx( B − A)=12c.E(Xn) =∫BAxn1⋅B −dxAn+1 n+1B − A=( n + 1)( B − A)137