# The Islamic University of Gaza Faculty of Commerce Economics

```The Islamic University of Gaza
Faculty of Commerce
Economics &amp; Applied Statistics Department
Mathematical stat. – Instructor : Ibrahem Abed
Nov. 12th, 2009 Time: 1 hour
Mid. Exam
Question #1:( 5 points)
let Y be a random variable with probability density function given by :
2 (1  y)
f ( y)  
 o
,
,
0  y 1
elsewhere
By using the method of transformation , Find the probability density function of U  2Y  1
Question #2:( 10 points)
a) If X denote a random sample with   60 and S 2  64
16
Find P( xi  992)
i 1
b) Suppose that  2  1.4 . Let S 2 denote the sample variance of the 20 observations
Find a number (a) such that P(a  S 2 )  0.975
Question #3:( 10 points)
a) How many observations should be included in the sample such that the sample mean will be
within 1 of the population mean , with probability 0.90 knowing that  2  16 ?
b) Let S12 denote the sample variance for a random sample of ten values and S 22 denote the
sample variance for the sample of eight values and S12 to be independent of S 22 such
that  1 2   2 2
Find a number b such that P(
S12
 b)  0.95
S 22
1
Question #4 :( 10 points)
Let Y1 ,Y2 be any two variables such that the joint density function of Y1 and Y2 given by
3 y
f ( y1 ; y 2 )   1
 o
,
0  y 2  y1  1
,
elsewhere
a) By using the method of Distribution Functions , Find the probability density function for U  Y1  Y2
b) Find E (U )
Question #5:( 5 points)
Let Y1 be a binomial random variable with n1 trails and probability of success given by p ,
Let Y2 be a binomial random variable with n 2 trails and probability of success given by p .
and Y1 ,Y2 are independent .
By using the method of moment generating functions,Find the density function for U  Y1  Y2
 02.025,20
 02.025,19
 02.025,18
 02.05,18
34.1696 32.8523 31.5264 28.8693
P(Z &gt; 2)
Z 0.05
Z 0.025
Z 0.10
0.0228
1.645
1.96
1.28
Z 0.02
F0.05,9 , 7
F0.05, 7 ,9
F0.05,10,8
F0.05,8,10
F0.10, 7 ,9
F0.10,9 , 7
P(Z &gt; 0)
2.05
3.68
3.29
3.35
3.07
2.51
2.72
0.5
P(Z &gt; 1)
df = 15
P(t &lt;1)
0.8334
df = 16
P(t &lt; 1)
0.8339
df = 14
P(t &lt; 1)
0.8328
0.1587
 02.975,20
 02.975,19
 02.975,18
 02.95,18
9.590
8.906
8.230
9.39
Good luck ☺☺☺
2
```