Discrete Distributions Chapter 5

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Discrete Distributions

Chapter 5


Random Variables

Random variablea variable (typically represented by x) that takes a numerical value by chance.

For each outcome of a procedure, x takes a certain value, but for different outcomes that value may be different.

pg. 205


Examples:

Number of boys in a randomly selected family with three children.

Possible values: x=0,1,2,3

The weight of a randomly selected person from a population.

Possible values: positive numbers, x>0


Examples:

Genetics: If two pea plants have both green and yellow pod genes, the probability that offspring pea plants has a green pod is .75

That is

P(green) = .75


Discrete and Continuous Random Variables

Discrete random variableEither a finite number of values or countable number of values (resulting from a counting process)

Continuous random variableInfinitely many values, and those values can be associated with measurements on a continuous scale without gaps or interruptions

pg. 206


Probability Distributions

Probability distributionA description that gives the probability for each value of the random variable; often expressed in the format of a table, graph, or formula

Pea pod histogram on page 207


x P(x)

0 1/8

1 3/8

23/8

3 1/8

Tables

Values: Probabilities:


GraphsThe probability histogram is very similar to a relative frequency histogram, but the vertical scale shows probabilities.


Requirements for Probability Distribution

∑ P(x) = 1 where x assumes all possible values.

0 P(x) 1 for every individual value of x.

pg. 207


Mean, Variance and Standard Deviation of a Probability Distribution

µ = ∑ [x • P(x)] Mean

σ 2 = ∑ [(x – µ)2 • P(x)] Variance

σ 2 = ∑ [x2

• P(x)] – µ 2 Variance (shortcut)

σ = ∑[x 2 • P(x)] – µ 2 Standard Deviation


• Press the STAT button and choose EDIT

• Enter the x-values into the list L1 and the P(x) values into the list L2

• Press the STAT button and choose CALC• Choose 1-Var Stats and press ENTER

• Type in L1 then , (comma) then L2 on that line, you will see 1-Var Stats L1,L2

• Press ENTER• You will see x-bar=…, it is actually m (mean)

and sx=…, it is actually s (st. deviation)

Using TI-83/84 calculator


Finding the Mean, Variance, and Standard Deviation

Example 5, page 209

Table 5-1 describes the probability distribution for the number of peas with green pods among five offspring peas obtained from parents both having the green/yellow pair of genes.


Using Excel - Mean

• Put the x-values in column 1

• Put the P(x)-values in column 2

• In column 3 enter “=A1*B1”

• Copy and paste that cell to the entire column of data

• At the bottom of column 3 enter “=sum(c1:cN)” - N is the last row of the data

Table 5-3, pg 209


Using Excel - Mean


Using Excel – Standard Deviation

Continuing

• In column 4 enter “=power(A1-$c$M,2)*B1”

• Copy and paste that cell to the entire column of data

• At the bottom of column 4 enter “=sum(d1:dN)” - N is the last row of the data


Using Excel – Standard Deviation


More detailed example

Using Excel to find the mean and standard deviation of a discrete

probability distribution


(1) Enter the values and their probabilities as separate columns

Using Excel to find the mean and standard deviation of a discrete

probability distribution


(2) Check to make sure the values of P(x) add up to 1:

Using Excel (continued)


(3) In column C, for each value of x, calculate x*P(x):



(4) The sum of column C will give the value of the mean, μ



(5) In column D, calculate the values of X2*P(x)



(6) Calculate the sum of the squares times probability - X2*P(x)



(7) Use the shortcut formula for the variance:



(8) Finally, the Standard deviation is just the square root of the variance:



How to Choose Lottery Numbers

• Note the sidebar on page 209 about choosing numbers in a lottery.


Round results by carrying one more decimal place than the number of decimal places used for the random variable x.

If the values of x are integers, round µ, σ, and σ2 to one decimal place.

Pg 210

Roundoff Rule for µ, σ, and σ

2


Identifying Unusual ResultsRange Rule of Thumb

According to the range rule of thumb, most values should lie within 2 standard deviations of the mean.

We can therefore identify “unusual” values by determining if they lie outside these limits:

Maximum usual value = μ + 2σ

Minimum usual value = μ – 2σ


Identifying Unusual ResultsBy Probabilities

Using Probabilities to Determine When Results Are Unusual:

Unusually high: a particular value x is unusually high if P(x or more) ≤ 0.05.

Unusually low: a particular value x is unusually low if P(x or fewer) ≤ 0.05.


Uniform Probability DistributionA uniform probability distribution has the following property:

P(x) = c

The value c is a constant, so every event is just a likely as every other event. If there are n events

P(x) = 1n


Binomial Probability DistributionA binomial probability distribution results from a procedure that meets all the following requirements:

1. The procedure has a fixed number of trials.

2. The trials must be independent. (The outcome of any individual trial doesn’t affect the probabilities in the other trials.)

3. Each trial must have all outcomes classified into two categories (commonly referred to as success and failure).

4. The probability of a success remains the same in all trials. pp 218-219


Notation for Binomial Probability Distributions

S and F (success and failure) denote the two possible categories of all outcomes; p and q denote the probabilities of S and F, respectively:

P(S) = p (p = probability of success)

P(F) = 1 – p = q (q = probability of failure)

Note that “success” may not be desirable.


Notation (continued)

n denotes the fixed number of trials.

x denotes a specific number of successes in n trials, so x can be any whole number between 0 and n, inclusive.

p denotes the probability of success in one of the n trials.

q denotes the probability of failure in one of the n trials.

P(x) denotes the probability of getting exactly x successes among the n trials.

pg 219


Methods for Finding Probabilities

We will now discuss three methods for finding the probabilities corresponding to the random variable x in a binomial distribution.


Method 1: Using the Binomial Probability Formula

P(x) = • px • qn-x (n – x )!x!

n !

for x = 0, 1, 2, . . ., nwhere

n = number of trials

x = number of successes among n trials

p = probability of success in any one trial

q = probability of failure in any one trial (q = 1 – p)

pg 221


Rationale for the Binomial Probability Formula

P(x) = • px • qn-xn ! (n – x )!x!

The number of outcomes with

exactly x successes among

n trials


Binomial Probability Formula

P(x) = • px • qn-xn ! (n – x )!x!

Number of outcomes with

exactly x successes among

n trials

The probability of x successes among n trials for any one

particular order


Method 2: Using TI-83/84

• Press 2nd VARS to get the DISTR menu

• Scroll down to binomialpdf( and press ENTER

• Type in three values: n, p, x (separated by commas) and close the parenthesis

• You see a line like binomialpdf(10,.3,6)

• Press ENTER and read the probability of the value x (successes) in n trials


Alternative use of TI-83/84• Press 2nd VARS to get the DISTR menu

• Scroll down to binomialcdf( and press ENTER

• Type in three values: n, p, x (separated by commas) and close the parenthesis

• You see a line like binomialcdf(10,.3,6)

• Press ENTER and read the combined probability of all values from 0 to x (i.e., probability that there are at most x successes)


Method 3: Excel

• Use an Excel canned function


(1) With the cell you want highlighted, From the home tab, click the arrow beside the ∑ button, and select “more functions”

Using Excel (2007 or later) for Binomial Distributions



(2) In the pop-up window, scroll down and select the category “statistical”



(3) Under “select a function” scroll down to BINOMDIST and click “ok”



(4) Fill in the values requiredNumber_s # of successesTrials

# of trialsProbability_s

probability of success Cumulative

TRUE if you want the probability of x or fewer successes, FALSE if you just want the probability of exactly x successes.



(5) Now look back at the cell you had highlighted:

From now on, you can just enter this short form if you find it quicker.


An “unfair coin” has a 0.55 probability of getting heads and is tossed 10 times

Example

• What is the probability of getting exactly 5 heads?

• What is the probability of getting at least 4 heads?


Probability of heads: p = 0.55

Number of tosses: n = 10 “Exactly 5 heads” → P(5)

P(5) = 0.234

Example


Probability of heads: p = 0.55

Number of tosses: n = 10 “at most 4 heads” → ∑P(4)

∑P(4) = 0.262

Example


An “unfair coin” has a 0.55 probability of getting heads and is tossed 10 times

Example

• What is the probability of getting exactly 5 heads?

• What is the probability of getting at least 4 heads?

p = 0.55 n = 10

P(5) = 0.234

∑P(4) = 0.262


Binomial Distribution: Formulas

Std. Dev. σ = n • p • q

Mean µ = n • p

Variance σ2= n • p • q

Where

n = number of fixed trials

p = probability of success in one of the n trials

q = probability of failure in one of the n trials


Interpretation of Results

Maximum usual values = µ + 2

Minimum usual values = µ – 2

It is especially important to interpret results. The range rule of thumb suggests that values are unusual if they lie outside of these limits:

Discrete Distributions Chapter 5

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