Discrete Probability
Distributions
WEEK FOUR
This worksheet relates to chapter five of the text
book (Statistics for Managers 4th Edition).
If you don’t understand something from
this week or last week please make sure
you ask because each week builds on the
preceding weeks.
Term
Expected value
of D.R.V.
Variance of
D.R.V
Mean of the
Binomial Distn
Std Deviation
of the Binomial
Distn
Covariance
Binomial
Formula
UNDERSTANDING THE RULES
Symbols
Definition
Weighted average over all possible outcomes.
E(X) = µ
What are the weights? Probabilities
2
Weighted average of the squared differences
σ
between each outcome and its mean.
E(X) = µ = np Have you looked at how to calculate binomial
probabilities on the computer?
σ
There is a good exercise on page 195 of your
text book. Have you tried plotting the
cumulative probabilities and histogram?
σXY
A measure of the direction and strength of
linear association between 2 random variables.
Random experiment involving repeated actions
where only 2 outcomes are possible
⎛n⎞ x
⎜⎜ ⎟⎟p (1 − p) n − x
⎝x⎠
⎛n⎞
n!
⎟⎟ and n C r (on calculator) and
are all equivalent.
X !(n − X )!
⎝X⎠
Please note: ⎜⎜
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DISCUSSION QUESTIONS
1.
What is a discrete random variable? What are some examples?
2.
What is a probability distribution for a discrete random variable? What
does it look like?
3.
Draw the binomial distributions for the following cases and say whether
they are symmetric, right or left skewed:
n=5 , p=0.1
n=5 , p=0.5
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n=5 , p=0.9
CALCULATION QUESTIONS
1. Using the binomial formula, find the probability of getting exactly one head
in three tosses of a fair coin?
Now check your result by drawing a tree diagram.
Make sure you use your formula sheet when
doing questions, because it is exactly the same
in the exam and it will help you if you know
where to look for your formulae. You will also
need to learn any formulae you need that are
not included on the sheet.
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2.
It has been a bad day for the market with 80% of securities losing value.
You are evaluating a portfolio of 15 securities and will assume a binomial
distribution for the number of securities that lost value.
Mid semester, April 2005
(a)
What assumptions are being made when you use a binomial
distribution in this way?
(b)
How many securities in your portfolio will you expect to lose
value? **
(c)
Find the probability that 10 securities lose value.
Want more practice?
Try the next one…
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3.
EXTRA PRACTICE: The records of an Italian shoe manufacturer show
that 10% of shoes made are defective. Assuming independence, find the
probability of getting:
(a) 2 defective shoes in a batch of 12.
(b) 6 defective shoes in a batch of 20.
MULTIPLE CHOICE PRACTICE
1. The binomial distribution is a _______ probability distribution
(a) categorical
(b) discrete
(c) continuous
2. Which of the following are not examples of a discrete variable?
(a) How tall you are
(b) Your age in years
(c) The number of cars you own
(d) The number of people in your class today
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3. The expected value for a random variable X is 20, and its variance is 49. The
expected value for a random variable Y is 30 and its variance is 64. The
covariance between X and Y is –13. What is the Var(X-Y)?
(a) 139
(b) 87
(c) 100
(d) 126
(e) 11
4. Which of the following statements is false for a binomial distribution?
(a) We are assuming sampling without replacement
(b) Only 2 possible outcomes for each trial
(c) The random variable is discrete
(d) Each trial is independent
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notes
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