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Discrete Probability Distributions

WEEK FOUR

This worksheet relates to chapter five of the text book (Statistics for Managers 4th Edition).

UNDERSTANDING THE RULES Term Symbols Definition

Expected value of D.R.V.

E(X) = µ Weighted average over all possible outcomes. What are the weights? Probabilities

Variance of D.R.V

σ2 Weighted average of the squared differences between each outcome and its mean.

Mean of the Binomial Distn

E(X) = µ = np Have you looked at how to calculate binomial probabilities on the computer?

Std Deviation of the Binomial Distn

σ There is a good exercise on page 195 of your text book. Have you tried plotting the cumulative probabilities and histogram?

Covariance

σXY A measure of the direction and strength of linear association between 2 random variables.

Binomial Formula

xnx )p(1pxn −−⎟⎟⎠

⎞⎜⎜⎝

⎛ Random experiment involving repeated actions where only 2 outcomes are possible

Please note: ⎟⎟⎠

⎞⎜⎜⎝

⎛Xn

and rnC (on calculator) and

)!(!!

XnXn−

are all equivalent.

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.

BRIEF SOLUTIONS

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DISCUSSION QUESTIONS 1. What is a discrete random variable? What are some examples?

Random variable: outcomes of an experiment expressed numerically. Eg. Toss a die twice and count the number of time the number 4 appears (0, 1 or 2 times). Discrete Random variable: a random variable that can only have certain distinct values. It is usually obtained by counting. Eg. Toss a coin five times and count the number of tails. (0, 1, 2, 3, 4 or 5)

2. What is a probability distribution for a discrete random variable? What

does it look like?

A probability distribution for a discrete random variable lists all the possible outcomes for the random variable together with the related probability

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 n=5 , p=0.9 right skewed symmetrical left skewed

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CALCULATION QUESTIONS 1. Using the binomial formula, find the probability of getting exactly one head

in three tosses of a fair coin?

n = 3 x = 1 p = 0.5 3C1 * 0.51 * (1-0.5)2 = 0.375

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? Assumptions of binomial distribution: 1. n observations in the sample 2. Observations are mutually exclusive and collectively exhaustive. 3. Probability of success is p an probability of failure is 1-p 4. All outcomes are independent i.e. if there is a finite population we are

sampling with replacement. 1. Constant probability of 0.8 applies to all securities 2. outcome of each security is independent of the other securities 3. identical trials = each security treated the same

(b) How many securities in your portfolio will you expect to lose

value? **

µ = np = 15 x 0.8 = 12 Expect 12 securities to lose value

(c) Find the probability that 10 securities lose value.

P(X=10 ⎢ p=0.8, n=15 ) = 15C10 x 0.810 x 0.25 = 0.103182

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. 0.2301 (b) 6 defective shoes in a batch of 20. 0.00887

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