CHAPTER 5 AND 6—FAMILIES OF RANDOM VARIABLES



Chapter 5 And 6--Families Of Random Variables.Doc

STATISTICS 301—APPLIED STATISTICS, Statistics for Engineers and Scientists, Walpole, Myers, Myers, and Ye, Prentice Hall

GOAL

In this section we investigate FAMILIES OF RANDOM VARIABLES. Many RV’s share similar properties and characteristics, so much so that we can define a “family” of such RV’s. Such families have a common definition of the RV and a common probability function or probability density function.

There are two classes of families of random variables, Discrete and Continuous. While there are many families of RV’s within each main class, we will only investigate the BINOMIAL and NORMAL families of RV’s. They are discrete and continuous, respectively.

In passing we will mention three other continuous random variables since they are important in statistical inference.

1. BINOMIAL DISTRIBUTION (or family of RV’s)

The Binomial family of RV’s is a set of discrete RV’s that all share the following in common:

1. The RV is defined the same.

2. The sample space is similar.

3. The probability function has the same “form.”

Consider the following three experiments.

| |I |II |III |

|Experiment |Toss a coin 4 times |Select 20 MU students at random |Select 10 people with cell phones at |

| | | |random |

|Facts |Coin is fair. |Known that 55% of MU students are |Known that Cincinnati Bell has 15% share |

| | |females. |of the cell phone market. |

|RV defn |H = # heads obtained |F = # females in our sample |C = # of Cinci Bell customers in our |

| | | |sample |

|Sample Space |SH = { } |SF = { } |SC = { } |

For each of the experiments, we can define a “sub-experiment” or a sub-piece of the entire overall experiment.

|Sub Experiment | | | |

Also note that in each case the RV is defined to be the number of successes, where a success is one of the possible “sub-experiment” outcomes.

|Success | | | |

We further note that the probability of a success is constant.

|Pr{ Success } | | | |

Lastly we note that the results on the different sub-experiments are independent.

Defn: A Binomial Experiment consists of an experiment in which:

1. there are “n” sub-experiments” or trials

2. each sub-experiment/trial results in one of two outcomes—called success/failure

3. in each sub-experiment/trial, the Pr{ success } is the same; call this probability “p”

4. the sub-experiment/trial outcomes are independent of one another.

Note that the three experiments above are Binomial Experiments. Binomial Experiments are defined by two constants: the number of trials and the probability of a success, and hence we differentiate between Binomial Experiment using these two values, n and p. Hence a Binomial Experiment with n and p would be denoted BIN EXP( n, p ).

The only restrictions on n and p in a Binomial Experiment are that:

1. n is a positive integer, so n ≥ 1

2. p is a probability, so that 0 ≤ p ≤ 1.

Defn: A Binomial RV is the NUMBER OF SUCCESS in a BINOMIAL EXPERIMENT.

Note that the RV’s defined above are all Binomial RV’s since in each case, the RV is the number of “successes” in a Binomial Experiment.

1. H = # of heads when a fair coin is tossed four times

2. F = # of females when 20 MU students are selected randomly

3. C = # of Cinci Bell cell phone customers among 10 randomly chosen individuals with cell phones

What’s different about the three different RV’s?

1. The NUMBER OF TRIALS/SUB-EXPERIMENTS is different for the three (4, 20, 10).

2. The probability of a success = Pr { Success }, call it p, is different ( 0.5, 0.55, 0.15).

Defn: A Probability Function for a Binomial RV X, where X is Bin ( n, p ) is given by

[pic]

where [pic] and recall that n! = n(n-1)…(2)(1) and 0! = 1.

Thus for the three Binomial random variables we have

1. H = # of heads when a fair coin is tossed four times = Bin ( 4, ½ ), so

[pic]

and the probability of 3 heads is Pr{H=3} = [pic]

2. F = # of females when 20 MU students are selected randomly = Bin (20, 0.55), so

[pic]

and the probability of 10 females is [pic]

3. C = # of Cinci Bell cell phone customers among 10 randomly chosen individuals with cell phones = Bin (10, 0.15) and

[pic]

and the probability of nine Cinci Bell customers is fC(9) = 0.0000003.

Example: Exercise 5.9 page 151

A truck tire is tested over a certain terrain and the US Army finds that 25% of trucks fail to complete the test run without a blowout (somewhere on the truck). Of the next 15 trucks that are tested find the probability that between 3 to 6 trucks will have a blowout.

Notes

1. Binomial RV’s are denoted as follows: if X is a Binomial RV, then we write X = Bin( n, p ), where n and p are the values that define the Binomial Experiment. As a result, the three random variables discussed previously are:

a. H = # of heads when a fair coin is tossed 4 times = Bin ( n = 4, p = ½ ) or Bin ( 4, ½ )

b. F = # of females when 20 MU students are selected randomly = Bin (20, 0.55)

c. C = # of Cinci Bell customers among 10 randomly chosen cell phone users =

Bin (10, 0.15).

2. Be careful of limits with the Discrete BINOMIAL RV. Suppose X is Bin ( 20, ½ )

Pr{ 3 ≤ X ≤ 6 } ≠ Pr{ 3 < X < 6 }!

since Pr{ 3 ≤ X ≤ 6 } = Pr{ X = 3, 4, 5, 6 } while Pr{ 3 < X < 6 } = Pr{ X = 4 or 5 }.

3. Suppose that X is Bin (n, p). We know that probability functions MUST sum to 1. Does the Binomials?

Recall the Binomial Theorem that says that [pic]

Since X is Bin ( n, p) we know that fX(x) =

Now let a = and b = and use the Binomial Theorem so that

4. Most texts have Binomial probability tables. Ours looks like

[pic]

and is one of many in the text. In fact, values for n range from 1 to 20 and values of p, as indicated in the above table range from 0.10 to 0.90, certainly not every possible value of p, but enough to give you answers for most of the HW problems.

5. TI-83 calculators have built in functions that will compute two kinds of Binomial probabilities. You can use a TI-83 to calculate individual probabilities of the form

Pr{ Bin( n, p ) = x} or, what are known as cumulative probabilities, of the form

Pr{ Bin( n, p ) ≤ x } = Pr { X = 0, 1, 2, …, or x }.

These functions can be found under the Stat [ Distn ] and are down the list at position “0” and “A”. The format of the function is as follows:

0: binompdf( n, p, x ) will give the individual probability, ie fX(x) = Pr{ X = x } and

A: binomcdf( n, p, x ) will give the cumulative probability, ie Pr{ X ≤ x }.

Recall that we calculated the probability of 10 males in our sample as 0.1594. Using a TI-83 we obtain

0: binompdf( 20, 0.55, 10 ) = Pr{ Bin(20, 0.55) = 10 } = 0.159349456 which, rounded, is 0.1594!

A: binomcdf( 7, 0.6, 4 ) = Pr{ Bin(7, 0.6) ≤ 4 } = 0.580096000

Defn: The parameters of a random variable are the constants in the probability function (or pdf is the RV is continuous) that can be different.

For the Binomial Family of RV’s, the probability function is of the form

[pic]

The parameters of the Binomial RV, Bin ( n, p ), are n and p. Think of parameters as a way of distinguishing different members of the Binomial family of RV’s. So a Bin (10, ½) is a Binomial RV but is different from a Bin (200, 0.005). Both share some common features:

1. they are the number of successes in a Binomial Experiment

2. both have probability functions of the form [pic], with different values of n and p,

3. both take on integer values between 0 to n.

THM: Let X be a Binomial random variable Bin ( n, p ). The mean and variance of X are

(X = n p σ2X = n p (1 – p).

We won’t prove this theorem in this class; you will see it’s proof in STA 401. Also, the mean and variance of every random variable will be a function of the parameters of the random variable!

Example #1—Exercise 5.9 page 151

Returning to the Army truck blowout example, we saw that B = # blowouts in the next 15 trucks that cover the test course is Bin ( n = 15, p = 0.25 ). We can then summarize the distribution of blowouts as follows:

the mean number of blowouts would be (B = n p = 15 (0.25) = 3.750

with a variance of σ2B = n p (1 – p) = 15(0.25)(1-0.25) = 2.8125

and a standard deviation of σB =1.6771.

The small standard deviation suggests that the distribution of values of B has rather small spread.

The Shape of the distribution is given by the Kind or Family of the random variable. So in this case we would state that the shape is Binomial.

Example #2 Exercise 5.5 page 150

E = number of operator errors among the next 20 failures = Bin ( n = 20 , p = 0.30 ).

a. Pr{ E ≥ 10 } = 1 – Pr{ E < 10 } = 1 – Pr{ E ≤ 9 }

= 1 – binomcdf(20, 0.30, 9) = 1 – 0.952038 = 0.047962

= 1 – 0.9520 = 0.0480

b. the probability of no more than 4 = Pr { E ≤ 4 } = binomcdf(20, 0.30, 4) = 0.23751

= 0.2375

c. you find 5 operator errors out of 20 or 5/20 = 0.25 which is “close” to the assumed probability of an operator error of 0.30! So 30% is reasonable!

We finally note that there are many other discrete families of random variables. At last count there are approximately six very common discrete random variables and many, many less common ones. Among the more common ones that you will encounter in other statistics classes are the

1. Discrete Uniform RV or Distribution

2. Geometric RV or Distribution

3. Negative Binomial RV or Distribution

4. Hypergeometric Distribution

5. Poisson Distribution

All of these distributions share a common feature: their sample spaces are either finite or countable!

II. Families of Continuous Random Variables

In this section we take a slightly different tack in defining random variables than we did for discrete random variables. For the most part discrete random variables can be defined by a physical experiment that will yield a particular kind of discrete random variable. The Binomial is an example. It was defined in terms of a Binomial Experiment. Continuous random variables, however, can not be so defined, so instead we define continuous random variable in terms of the probability density functions.

1. The Normal Distribution

The Normal family of random variables or distribution is the most important distribution in statistics. Most statistical inference methods assume a Normal distribution. Further, many other continuous families of random variables are “generated” by the Normal distribution.

For the continuous random variables that we will investigate, we will employ the following method. We will first define the RV in terms of its pdf. We will then investigate various features of the random variable such as its sample space, its parameters, how the shape of the pdf is determined by the parameters, how to calculate probabilities for it.

Defn: A Normal Random Variable or Distribution is any continuous RV with pdf of the form

[pic]

Notes

1. The Normal Distribution pdf looks like:

2. What are the parameters of the Normal and what do they correspond to?

3. How will we denote a Normal RV?

4. What is the Pr { 4 < X < 10 }?

5. Here are three examples of Normal Distributions. Describe each one.

[pic]

Normal Distributions are denoted by N( (, ( ). This is shorthand for saying that a distribution of numbers has a Normal Shape (the N), whose peak occurs at the value of (, and whose spread is given by (. For example, in the US the distribution of heights of adult females is approximately Normal with a mean of 56.5 inches with a standard deviation of 8 inches. Using shorthand we would say, “Females heights are N(56.5, 82=64).” Pictorially the distribution of female heights is approximated by the following figure.

[pic]

Probabilities for Normal Distributions can be found a couple of ways. First there is a rule, called the “Empirical Rule” that states:

”For Normally Distributed populations:

a. 68% of the population falls within one standard deviation of the mean,

ie [ ( - 1 (, ( + 1*( ].

b. 95% falls within two standard deviations of the mean, and

c. 99.7% falls within three standard deviations of the mean.

More Notes concerning the Normal Distribution

1. A Normal Distribution’s mean can be positive, negative, or zero. The standard deviation can never be negative and hence must be greater than or equal to zero.

2. A special Normal Distribution is used in practice and is known as a Standard Normal. A Standard Normal is a Normal Distribution with a mean of 0 and a standard deviation

of 1. We usually use Z to denote the Standard Normal, N(0,1).

3. The Standard Normal Table is a table with the probabilities of Z given for various values of Pr{ Z < z } for values of z from –3.49 to +3.49.

|[pic] |[pic] |

For example:

Pr { Z < -2 } =

Pr { Z < +2 } =

Pr { Z > +2 } =

Pr { -2 < Z < +2 } =

More Notes concerning the Normal Distribution (Continued)

4. Any Normal Distribution can be converted to a Standard Normal by “standardizing” it. To standardize a number, you subtract the mean, and divide the result by the standard deviation. That is, to standardize a N((, σ2) we [pic]. For example, we saw above that US adult female heights are N(56.5, 82). If we were to take all the heights, subtract off 56.5, and then divide by 8, [pic], the resultant distribution of numbers would have a Standard Normal, N(0,1), distribution.

5. Probabilities for Normal Distributions can be found using the Standard Normal Tables as follows:

| | |

In general, any Normal RV can be standardized into a Standard Normal!

For example, suppose X = N( 10, 4). Find the following probabilities using the Standard Normal Distribution.

|[pic] |[pic] |

For example:

Pr { X < 8 } =

Pr { 8 < X < 14 } =

More Notes concerning the Normal Distribution (Continued)

6. In the section on Boxplot, outliers were defined as any observation “outside” the fences of the Boxplot. Outliers are more commonly defined in terms of their “Z-score.”

That is, an outlier is typically defined to be any value, more than three standard deviations from the center (average or mean). Since a Z-score is the number of standard deviations an observation is distant from the mean, an observation with a z-score less than -3 or greater than +3 is usually deemed an outlier.

The basis for this argument has to do with the probability of such a value. Using the Standard Normal Table, we notice that the chances of an observation falling more than 3 standard deviations away from the mean, which is equivalent to saying it’s z-score is beyond ( 3, is 0.003, 0.3%, or only [pic]. In short, it is VERY UNLIKELY, we should get values more than three standard deviations away from the mean, IF THE DATA ARE NORMALLY DISTRIBUTED.

IF, HOWEVER, THE DATA ARE NOT NORMALLY DISTRIBUTED, then all bets are off and such extreme values could be VERY LIKELY!

7. As with any distribution, values near the center of Normal Distributions are more likely to occur compared to observations “out” in the tails of the distributions, way far away from the center, measured in how many standard deviations away from the center.

That is, suppose we select an observation from a Normal Distribution, it is most likely that the observation will be near ( rather than far away. While a value far away from the mean COULD OCCUR, it is unlikely, while a value near the center is much more likely.

8. The TI-83 also has the capability of finding Normal probabilities. Under the Distn menu you have:

1: normalpdf(x [, (, σ ] ) gives the density values, ie the height of the density and

2: normalcdf(lower, upper [, (, σ ] ) gives Pr{ lower < N((, σ) < upper }

More Notes concerning the Normal Distribution (Continued)

8. Lastly we introduce some common notation that we will encounter on a regular basis.

Defn: The value z α represents the Standard Normal value such that α of the probability (or α of the area) of the Standard Normal Distribution lies to the right of this value. That is:

Pr{ Z > z α } = α.

[pic]

Find the following values:

z 0.10

z 0.05

z 0.025

We finally present three other important continuous distributions or random variables that will occur frequently in the statistical inference methods we will see later. Rather than go in great detail we will only briefly investigate them, focusing on:

1. their density function form and shape

2. the parameters of the distribution and values they can take on

3 the mean and variance of the distribution

4. how to find probabilities for these distributions.

2. The Chi-Square Distribution

Defn: A Chi-Square Random Variable or Distribution is any continuous RV with pdf of the form

[pic]

where [pic] and ((n) = (n-1)! for n an integer.

Chi-Square Distribution Notes

1. The Chi-Square Distribution pdf looks like:

Not symmetric!

2. What are the parameter(s) of the Chi-Square and what does (do) it(they) correspond to?

The parameter is more referred to as the degrees of freedom or df for short.

3. We denote a Chi-Square random variable or distribution by

4. The mean and variance of the [pic] are df and 2df, respectively.

Chi-Square Distribution Notes (Continued)

5. Most texts will have tables with probabilities for the Chi-Square distribution. Our text’s tables looks like

|[pic] |[pic] |

Example: Let X be a Chi-Square RV with 10 df’s.

Pr{ X ≥ 2.558 } =

Pr{ X < 2.558 } =

6. Upper Percentiles of the Chi-Square Distribution

Defn: The value (2(α, df) represents the value in a Chi-Square distribution with parameter, df, such that α of the probability (or α of the area) lies to the right of this value. That is:

Pr{ (2(df) > (2(α; df) } = α.

(2(0.05, 30) =

7. Relationship to the Normal

The Normal Distribution is so central to all of statistics because most other distributions are defined in terms of some function of a Normal.

The Chi-Square distribution is defined in terms of the Normal distribution as follows:

A Chi-Square random variable or distribution with k degrees of freedom is the sum of k independent Standard Normals squared. That is:

(2(k) = [pic]

8. The TI-83 also has the capability of finding probabilities for the Chi-Square distribution. Under the Distn menu you have:

6: (2pdf(x , df ) gives the density values, ie the height of the density and

7: (2cdf(lower, upper , df ) gives Pr{ lower < (2 (df) < upper }

3. The Student T Distribution

Defn: A Student T Random Variable or Distribution is any continuous RV with pdf of the form

[pic]

where [pic] and ((n) = (n-1)! for n an integer.

T Distribution Notes

1. The T Distribution pdf looks like:

and looks VERY SIMILAR TO A STANDARD NORMAL Distribution!

2. What are the parameter(s) of the T Distn and what does (do) it(they) correspond to?

The parameter is also referred to as the degrees of freedom or df for short.

3. We denote a T random variable or distribution by T(df).

4. The mean and variance of the T(df) are 0 and a number > 1, respectively, because???

5. In the limit, as df ( ∞, a T(df) ( Z or N(0,1). So as df gets large for a T distribution it approaches the Standard Normal Distribution and hence could be used to approximate the T with large df’s!

T Distribution Notes (Continued)

6. Most texts will have tables with probabilities for the T distribution. Our text’s tables looks like

|[pic] |[pic] |

Example: Let X be a T random variable with 10 df’s or X = T(10).

Pr{ T(10) ≥ 2.764 } =

Pr{ T(10) < -2.764 } =

Pr{ -2.764 < T(10) < 2.764} =

T Distribution Notes (Continued)

7. Upper Percentiles of the T Distribution

Defn: The value t(α, df) represents the value in a T distribution with parameter, df, such that α of the probability (or α of the area) lies to the right of this value. That is:

Pr{ T(df) > t(α; df) } = α.

t(0.05, 10) =

8. Relationship to the Normal

The T distribution is defined in terms of the Normal distribution as follows:

A T random variable or distribution with k degrees of freedom is the ratio of an independent Standard Normal the square root of a Chi-Square random variable (with k degrees of freedom) divided by its degrees of freedom. That is:

T(k) = [pic]

9. The TI-83 also has the capability of finding probabilities for the T distribution. Under the Distn menu you have:

4: tpdf(x, df ) gives the density values, ie the height of the density and

5: tcdf(lower, upper , df ) gives Pr{ lower < T(df) < upper }

3. The F Distribution

Defn: A F Random Variable or Distribution is any continuous RV with pdf of the form

[pic]

where [pic] and ((n) = (n-1)! for n an integer.

F Distribution Notes

1. The F Distribution pdf looks like:

Not symmetric!

2. What are the parameter(s) of the F and what does (do) it(they) correspond to?

The parameter is also referred to as the degrees of freedom 1

and degrees of freedom 2 or df1 and df2 for short.

3. We denote an F distribution by F(df1, df2).

F Distribution Notes (Continued)

4. Most texts will have tables with probabilities for the F distribution. Our text’s tables look like where a set of tables have a certain probability attached to it; usually these probabilities are 0.10, 0.05, 0.025, and 0.01. Our text gives us just the 0.05 and 0.01 tables.

|[pic] |[pic] |

Example: Let X be a F random variable with the following df’s.

Pr{ F(2, 10) ≥ 4.10 } =

Pr{ F(10, 2) > 19.40 } =

So note that the order of the df’s is crucial! Changing the order will yield a DIFFERENT RANDOM VARIABLE!

F Distribution Notes (Continued)

5. Upper Percentiles of the F Distribution

Defn: The value f(α, df1, df2) represents the value in a F distribution with parameters, df1 and df2, such that α of the probability (or α of the area) lies to the right of this value. That is:

Pr{ F(df1, df2) > f(α; df1, df2) } = α.

f(0.05; 2, 10) =

f(0.05; 10, 2) =

7. Relationship to the Normal

The F distribution is defined in terms of the Normal distribution as follows:

An F random variable or distribution with df1 and df2 degrees of freedom is the ratio of two independent Chi-Square random variables divided by their degrees of freedom. That is:

F(df1, df2) = [pic]

8. The TI-83 also has the capability of finding probabilities for the F distribution. Under the Distn menu you have:

8: Fpdf(x, df1, df2 ) gives the density values, ie the height of the density and

9: Fcdf(lower, upper , df1, df2 ) gives Pr{ lower < F(df1, df2) < upper }

-----------------------

Continuous

[pic]

[pic]

[pic]

Discrete

All Random Variables

[pic]

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