Question

2. Suppose that X1, ..., Xd N(0,9) and suppose that Y1, ..., Y10 d (1,9). Assume that all the Xs and Ys are independent of one another. (a) Determine the distributions of X and Y. (b) Determine the distribution of X1/Sy. (c) Compute P(S< 2.393). (d) Find an interval (a, b) such that Pla < s <b) = 0.95. Make the interval such that s has equal probability of being below a as of being above b. (e) Determine the distribution of (1) Compute P > 7.96). (g) Find an constant b such that P < b) = 0.90. (h) Determine the distribution of X-Y.

Answer 1

(a)

É ~ N(0,9)

Y ~ N(1,2)

b) This is nothing but a standard normal variate and as variance of X and Y are same. So, one can use .

Sy or Sy

X1-0 Sy ~ N(0,1)

c)

P(S} < 2.393)=P( <5*2.393) = P(x) < 1.329444) = 0.06812705

d)

$P(a< S_X ^2 < b) = P(\frac{a }{9} <\frac{5 S_X ^2}{9} < \frac{b }{9} ) = P( \frac{a }{9} <\chi^2_{(5)} < \frac{b}{9} ) = 0.95$

0.95 Po <xt)) = 0,95 = 0.47 -= 0.475 g = 4.171484 a = 37.54336

P> x) = 0 = 0.475 = 4.536611 b = 40.8295

e)

F5.10 (5,10)