Question

6: Suppose the random variable X has the uniform distribution on [a,b]. Find expression involving a and b for the expected value, variance, and standard deviation of X. Check that you expressions when a = 0 and b = 10 agree with what you got in part c) of problem 5.

Answer 1

| Pago No Dato then p. d. f. of x xouca,b) f(x) = 1 is 5 – a gxcЬ expectation of x , E(X) = Sx. fra da Bois b = (x. Ad xo b-a ао асор = (x.da - ba a 2 7 ore 2 (b-a) (b-d) (bta oor 2 (b-a). TEX= ato - atbT 22 VO) - EcX2) - [EO]? E(X²) = 3x² from da 8 2 1 da a adx I dx = t (x bral b- a

Pago No. Dato E(X²) = I. (6²-a3 8(b-a) To (bra (6²ab + a2) 3Cbra EC?)= (b + ab +a%). 3 ~(X) = 6²7 abta² rab = 6²+ abta - Calthab +6²) 3 hogy - 46+4 abt 4 a? 30 6ab-3b" b'-zab+a? IyCX = r icab) a zabt be (b-a) T 12 standard deviation (S.D.) S . = VOY Cb- ? (b- a 443 2 IS. I = (b-a) 2 3

Pago No. Dato ie. G ,b=10 YaU(Oilo) j C L10 10- |E() = 3t = Te - | VID = b-- Co-o, 100 10 12 IvD = 83333| SD= -) = 10 - 2.21 2XA 2 | 130. - 2.2267)