1) A water valve controls the amount of water flowing through the Beiber dam. The probability for the
water flow is uniform between 0 and 1 b/s (barrels per second), uniform between 1 and 2 b/s, and
uniform between 2 and 3 b/s. The probability of being between 1 and 2 b/s is half as likely as the
probability of being between 0 and 1 b/s, but twice as likely as being between 2 and 3 b/s. The water
flow cannot be 3 b/s or greater. In other words, the pdf look like this:
 k
0  x 1

f ( x)   1 k 1  x  2
2
1
 4 k 2  x  3
What value for k would make this a valid pdf?
4/7 = .5714
2)
Resistors A research group of engineers has determined that the probability density curve for the
proportion of resistors that survive the first test:
f(x) = k*(x2 – 2x + 1) for 0 < x < 1
but they need to know the value of k that makes this a valid probability density.
What is the value of k?
3
3) The time for the first widget to be manufactured each morning is random between 1 and 3 s with
f ( x)  15 x - 3 x - 1
16
2
3
for 1  x  3
a) What is the cumulative distribution function, F(x)?
1/32(5x^6-54x^5+225x^4-460x^3+495x^2-270x+59)
Or if they factor
1/32(x-1)^4(5x^2-34x+59)
b) What is the probability of a widget being made earlier than 2 s? 0.34375
c) What is the probability of a widget being made later than 2.5 s? 0.1694
d) What is the probability of a widget being made earlier than 2.71 s? 0.9567
4) The amount of weight that the Elephant Hook® is required to hold is random depending on what is
hung on it. It has to hold at least it’s own weight of 10 grams, and if the weight is over 70 grams it will
pull out of the sheetrock. This means the pdf for the Elephant Hook® is
f(x) = (x-10)/1800 for 10 < x < 70
a) What is the expected value, μx, for the weight on the Elephant Hook®? 50
b) What is the variance, σ2x, for the weight on the Elephant Hook®? 200
c) What is the standard deviation for the weight on the Elephant Hook®? 14.14
d) What is the median for the weight on the Elephant Hook®? 52.426 (If you integrate from 0 you
get 53.59)
e) What is the probability a random Elephant Hook® will need to hold more than 50 grams?
5/9 = 0.56
f) What is the probability a random Elephant Hook® will need to hold 50 grams or more?
5/9 = 0.56
5) The Elo Chess Rating System is a method for predicting scores from a chess match (in this case under
a certain scenario). The probability of getting a certain score (for any real number x) is
f ( x) 
e  ( x1500) / 500
1  e ( x1500) / 500
The cumulative distribution function (for any real number x) is
F ( x) 
a)
b)
c)
d)
e)
1
1 e
( x 1500) / 500
What is the probability of getting a score of 1600? 0
What is the probability of getting a score less than 1600? 0.55
What is the probability of getting a score greater than 1800? 0.35
What is the probability of getting a score between 1600 and 1800? 0.10
Write the equation you would use to find the variance of the scores.



x2
  2 e  ( x1500) / 500

e  ( x1500) / 500
  x
dx

dx  2
( x 1500) / 500

(
x

1500
)
/
500
1 e
  1  e
