Question

all questions are from last year exam paper.I need solution for all of them so that I can prepare for my exam.Please help

(b) Let the random variable X with the probability density function f(x) = 2x; 0<x<1. Find: the p.d.f. of Y = 8X3 (i) (ii) E(Y) and Var(Y). Also show that E(Y) = E(8X3). 6 4. (a) The probability that a patient recovers from a delicate heart operation is 0-8. What is the probability that exactly 2 of the next three patients who have this eperation survive? (j) (ii) all of the next three patients who have this operation do not survive? (ii) all of the next three patients who have this operation survive? 4 Develop a test for testing (b) against H1 : fix) = 1,0 < x < 1 based on a sample of size n = 1 . Also find an expression for size and power of the test. MTE-11 P.T.O.

Answer 1

The PDF is $f_X(x)=2x;0<x<1$ .

i) The PDF of the transformation $Y=g(X)$ is

dy

Here

Y1/3 2

1/3 1/3 | У-2/3 I 2 6 1/3'

ii)

The expected value is

r 8 0 8 ду o 6y1/3 3y5/318 5 6 Jo 16

r8 0 y2 ду 85/3 0 E(Y2)- 16

The variance is

Var(Y) E(Y)- E(Y) 16,2 Var(Y) 16-5 144 Var(Y)25

Now,

0 .1 0 E(8x*)-16-E(Y) 5