Question

9. (14 points) Let X be a continuous random variable with probability density function Vix { Ook 1 2. otherwise (a) Sketch of the density function. Indicate P(x>) in your sketch. (b) Find P(X >). (e) Find the expected value, E(X). (d) Find P(X < _X >)).

Answer 1

be the Vandom Vavable Vanable of interest. The of interest. The pdf of a s given by, fx cx ) 20 Ο Ζα Ζ) 1 2-2 14227 0.w (a) Sketch the density function. Indicate P(x > 2/2 in your Sketch. sketch fx 2 h (111) 2-7 P(x). at anpa 2 (6) Find P (x>* 2 Since 12 3 2 La Hence P(x? 3) = frx) dx G je-2342 rold 2

23- 20 0.125 (©) Expected value E (x) com be defined an, Ecx)= s x-fox) dx = 2 x 2 dx+ x(2-x)da 2 3. + (22-23 dx V 2 2 + 23 ). 3 글 + (4 4-5)-(1-] 0.66661 (4) Р(х4 3 | *>) - PGL X<) P (x² / 2 3/2 * ( * <** ). [fcaudz + (fex)de P (312) 11 š xdx + Je-z)dx |(2-x)dx= 0.375+ 0.315 - 0.75

3- 2 P(x) ad act (2-x) dx -x) dx = 0.375+ 0.5 = 0.875 Substitating the values to (O Сое get, P(xc min |x>) P ( 1 / 2 Lxc 32 0 75 11 P(x) 0.875 0.8571