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Page 1
Exam 2 Introductory Statistics for Engineers
Fall 2014
Name ________________________
Each Problem is worth 50 points.
1. The following distribution (cumulative probability distribution) function F(x) gives the
probability that the value of the random variable is less than or equal to x.
if x  0
 0
 x

if 0  x  1
 4
F  x  
 3 x  1 if 1  x  2
4 2
 1
if x  2

a)
Determine the probability density function f(x)
b)
Determine the expected or expectation value of x: E  x   x
/150
 
c)
Determine the expected or expectation value of x2: E x 2  x 2
d)
Determine  x2 and  x
f ( x) 
x  x 
__________________
x2 
__________________
 x2 
__________________
x 
__________________
e)
Compute the probability that the random variable takes on a value between
f)
Compute the median score of this distribution.
1
3
and .
2
2
Page 2
Exam 2 Introductory Statistics for Engineers
Fall 2014
2. The joint probability density function for two random variables x and y is given by
 x  y if 0  x  1 and 0  y  1
f  x, y   
.
elsewhere
 0
Determine the probability density functions of x, f1 ( x ) and y, f 2 ( y ) . Determine the mean and
standard deviation of x and the mean and standard deviation of y.
f1 ( x ) 
f2 ( y) 
x 
__________________
y 
__________________
x 
__________________
y 
__________________
cov  x, y  
__________________
  x, y  
__________________
Compute the Expectation value of w  14 x  2 y  3 .
w  ________________________________
Is the population variance of w  14 x  2 y  3 ,  w2 , equal to 196 x2  4 2y ? Explain your answer.
3. A cut piece of pitch pine is going to be used as a support beam. The density of the wood is not
uniform throughout its interior, having a mean value of 0.674 g/cm3 and a standard deviation of
0.027 g/cm3. Density measurements are made at 40 randomly chosen core samples of the piece.
a) Calculate the probability that the sample mean of the 40 density measurements is between the
values 0.668 g/cm3 and 0.678 g/cm3.
b) 75% of the time the mean density of these 40 core samples would be smaller than what value?