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Chapter 5-3

MGMT 4800 Chapter Notes - Chapter 5-3: Probability Distribution, Negative Number, Standard DeviationExam


Department
Management
Course Code
MGMT 4800
Professor
Joliuna
Chapter
5-3

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STAT 1430 REC 15(SOLUTIONS) SEC 4.3
CONTINUOUS R.V.
For every question in this recitation assignment, assume X is a continuous random variable
with a probability density function). Draw pictures where appropriate and helpful.
1. Which of the following can be a possible set of values for X when X is a continuous RV?
a. X = 0, 1, 2, 3, …
b. X greater than equal to 0
c. Both b and c are correct
2. What are the two conditions that are required in order for f(x) to be a legitimate
probability density function?
i. f(x) 0
ii. 
3. We find a probability for a continuous random variable X by finding the AREA under
the CURVE.
4. If X is continuous, then f(2) stands for the probability that X = 2.
a. True
b. False
5. When X is a continuous random variable, P(X < 4) = P( X 4)
a. True
b. False
6. Which of the following could possibly be a negative number?
a. f(x)
b. p(x)
c. the set of possible values for X
d. None of the above.
7. Let X represent the amount of time students use when they take a final exam. The allotted
time is up to 2 hours. Suppose X is continuous and its pdf is below.
f(x) = (3/8)x2 where 0 < x < 2 (in hours)
a. Show that f(x) is a legitimate pdf (2 conditions)
i. , for all 0 < x < 2
ii.


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STAT 1430 REC 15(SOLUTIONS) SEC 4.3
CONTINUOUS R.V.
b. Graph:
c. Find P(X < 1).


d. Without doing another integral, what is P(X > 1)?
P(X > 1) = 1 P(X < 1) = 0.875
e. Find the MEAN amount of time students use to take the exam. Show all work.





 hrs
f. Find the MEDIAN amount of time students use to take the exam.
We want to solve:

Then


This is the same as solving
 hrs
g. Why can’t we use geometry in this problem?
Because we don’t have a simple pdf (like a rectangle) that we can find areas of by
known equations (like base times height).
h. SET UP BUT DO NOT CALCULATE the integral that finds us the variance of the
amount of time students use to take the exam. Be sure to include the limits of
integration.

 =



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STAT 1430 REC 15(SOLUTIONS) SEC 4.3
CONTINUOUS R.V.
i. To find the standard deviation, what do you do with the previous result?
Take the square root:


8. A fast food manager wants to be prepared to have enough of COIN change (pennies, nickels,
dimes, and quarters) at the drive through window ready to give to customers. Let X be the
amount of COIN change that any customer is given back when they pay for their food at the
drive up window (in decimal form.) Note: The amount of coin change would be anywhere from
0 cents to 99 cents = .99 dollars. Suppose f(x) is below.
f(x) = 6x(1-x) where 0 ≤ x < 1.00 (in decimal form)
a) Find the value of k that makes this a legitimate pdf.
All pdfs integrate to 1. Therefore we need to solve the following equation for k:
 

The integral equals
 

Therefore, .
b) Graph f(x)
c) Show that f(x) is a legitimate pdf for X.
i. f(x)≥0 for all x from 0 to 1
ii. 


d) Find the mean amount of change a customer receives back when paying for their order.
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