Probability and Statistics Test Set 8

1. The life of a pencil cell is a random variable with mean 60 hrs and standard

deviation 15 hrs. A cell is used until it fails, at which point it is replaced by a new one. Assuming 20 such cells are available with independent lives find the approximate probability (using central limit theorem) that over 1300 hrs of use can be obtained.

2. Let X, the lives of electric coil used in heater have mean and s.d. 120 hours. n of these

coils are put on test till they fail, resulting in observations 1 nX ,..., X find the minimum

value of n so that the probability that X differs by by less than 60 hours is at least 0.95?

3. Let X1, , X11 be i.i.d. N(, 2), find P( 11 X 1.796 S). 4. Let X1, , Xn, Xn+1, Xn+2 be i.i.d. N(, 2) and let X and S2 denote the sample mean

and sample variance based on X1, , Xn. Find the distribution of

n 1 n 2n X X 2X2(n 2) S

. 5. Let X1, , Xm be a random sample from N(1, 2) population and Y1, , Yn be another

independent random sample from N(2, 2) population. Let 2221 S,S,Y,X be the sample means and sample variances based on X and Y-samples respectively. Determine the distribution of

1 22(X ) 3(Y )U

S 4 m 9 n , where

2 2x y2 (m 1)S (n 1)SS .

(m n 2)

6. Consider two independent samples- the first of size 12 from a normal population with

variance 6 and the second of size 6 from a normal population with variance 3. Compute the probability that the sample variance from the second sample exceeds the one from the first.

7. The temperature at which certain chemical reaction takes place is normally distributed

with variance 2. A random sample of size n has the sample variance S2. How many observations are required to ensure that P(S2/2 1.8) 0.99?

8. Let X1, X2, X3, X4 be independent N(0, 2) random variables. Find

) . 2 2 2 2 21 2 3 4P(X X X X

9. Let X1, , Xn be a random sample from N(, 2) population. For 1 < k < n, define .VX

1kn1T,UX

1k1S,X

kn1V,X

k1U

n

1ki

2i

k

1i

22i

2n

1kii

k

1ii

Find the distributions of 2 2 2

1 2 3 42 2

5 62

(k 1)S (n k 1)T S k(U )W U V,W ,W ,WT T

(n k)(V ) k(n k)W and W (U V).T n(n 2)W

,