Question

3. Let X be a continuous random variable with probability density function ax2 + bx f(0) = -{ { for 0 < x <1 otherwise 0 where a and b are constants. If E(X) = 0.75, find a, b, and Var(X). 4. Show that an exponential random variable is memoryless. That is, if X is exponential with parameter > 0, then P(X > s+t | X > s) = P(X > t) for s,t> 0 Hint: see example 5.1 in the book.

Answer 1

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and o S'fres du = 1 s'entraining a s'ox?r for o [2 ? JO + b[2] = 1 = [1-0] + [1-0) - ) 2 a + محال لم dx @ 29 + 3b =) r 6 I 29+3ba6 ENG and can ( vo 201 to 3b = 6 from eano 3a + 4b = log zatott 4b CSscanned with Cancerinel ply by D can ☺ by 3 and eqho by z

6 a tgb = IS 69 a 8 b 15 b put b=0 in con 1 29 + 3XO 2a = 6 al 3 hare b=0 To find out rar (x) a =3, ECY?) = Ş x f(x) dx Sx caxbx x cax²tbx dx put a= 3, b=o x? [3x? 3x +o] da. 3x4 dx. 3a [] 3 [- 3 CSScanned with CamScanner S

E 3 ܗܝ ܐܟ E Cx2) - 06 yar(x) = E(+2) - [E(x)] 2 016-(075 vax(x) = 0.0375 4) То show an exponentie rondom yorlable is memoryless iie P[*> 5+{\x>s] = P(x>] pas → YN Exp () P[x>, 2= 1- Fla) = L.H.S. - p[x75+\>s] - PT(*>5++),(*))] PE*») e-xx P[X2,5+) - (stt - e P[x>s ents ext R.H.S - P(x > {] e-at R.His a l. His CSscanned with Campsp [>s+t\*>s] = p [x>]