Random Variables In this chapter we will continue our study of building probability models for random phenomenon's. In particular we will examine those.

Random Variables

In this chapter we will continue our study of building probability models for random phenomenon's.

In particular we will examine those random phenomenon's whose outcomes can be expressed as numerical values. By the end of this chapter you should be familiar with different types of discrete random probability models and continuous random probability models and how to find the means and variances of these models.

Review Definition of a Probability Model

There are two things we must identify when building a probability model for a random phenomenon. What are they?

1)

2)

Random variable

• The outcomes of the sample space are not always numeric. For instance flipping a coin 4 times could give the string of HHHT or HHTT.

• In statistics we are most often interested in numerical outcomes of a random phenomenon such as the count of heads in four tosses.

• Because of this we might assign values to the outcomes of the random phenomenon (This is only for those situations that it makes sense to do so) The outcomes of the sample space are now represent by a variable X ( a set of numeric values) called a random variable.

• Definition:• Random Variable: Is a variable whose value is a numerical outcome

of a random phenomenon.

Random Variable

• As we progress from general rules of probability toward statistical inference, we will concentrate on random variables. When a random variable X describes a random phenomenon, the sample space S just lists the possible values of the random variable. We usually will not list S (sample space) separately.

Assigning Probabilities to the values of a random variable.

• There are two types of random variables (thus two types of probability models) that will dominate our application of probability to statistical inference.

• Each will require different techniques for assigning probabilities to the values the random variable represents. Remember assigning the probabilities to the outcomes is the second step in building a probability model for a random phenomenon.

Discrete random variable• A discrete random variable X has a countable number of possible

values. The probability distribution of X lists the values and their probabilities.

• The probabilities must satisfy two requirements. What are they?• 1)

• 2)

• Find the probability of any event by adding the probabilities p of the particular x values that make up the event.

1 2 3 k

1 2 3 k

of X x x x .......... x

: p p p .......... p

Value

probability

Examples:

Let’s create a probability distribution of the random phenomenon of flipping a fair coin 4 times and counts the number of heads that occur. This is a discrete random variable we will call X.

What assumptions must we take into consideration with this model?

• Assumptions:• 1)

• 2)

Example:

The count of the number of heads of 4 tosses of a fair coin.

Outcome random variable probabilityHHHH 4 1/16HHHT 3 1/16HHTH 3 1/16HTHH 3 1/16THHH 3 1/16TTTH 1 1/16TTHT 1 1/16THTT 1 1/16HTTT 1 1/16HHTT 2 1/16TTHH 2 1/16THHT 2 1/16HTTH 2 1/16THTH 2 1/16HTHT 2 1/16TTTT 0 1/16

Probability distribution of X the number of heads in 4 tosses of a fair coin.

• What is the set X, the number of heads we can obtain when we flip a coin four times?

• X = { } remember this will now represent our sample space for this event.

• Why is X called a discrete random variable?

• Now assign probabilities for each value of set X. Are they equally likely for each value of the discrete random variable X? Give these as relative frequencies

Number of heads in four tosses of a coin probability model continued

Outcome Random variable X P(X)

4 heads 4 1/16

3 heads 3 4/16

2heads 2 6/16

1head 1 4/16

0 heads 0 1/16

Probability distribution for X

• X= the number of heads in 4 tosses of a fair coin. • X 0 1 2 3 4• P(X) .0625 0.25 0.37 0.25 .0625

• BIG POINT: This is a probability model. It is a theoretical model. The probabilities can be thought as the relative frequencies of the random variable if I did conducted the experiment and infinite number of times. A probability model is still based on assumptions. The assumption in this model is that a head and tail are equally likely. We obtained this assumption from the people that flipped the coin 40,000 times. (heroically)

Probability histogram

• Is a histogram of a probability model of a discrete random variable.

• How would we draw it?

Number of heads in 4 tosses

Relative frequency

What does the area of each bar represent?

Relative frequency model.• A relative frequency model can help estimate a

probability model if the experiment is conducted many, many, many, times!

• The probability model is the theoretical model and the relative frequency model is what actually did happen. One idea of statistical inference is to compare what did happen (relative frequency model) to what we EXPECT to happen (probability model) and draw conclusions.

Example:

• What is the probability of winning on the first roll in the game of craps? To answer this question we will build a probability model of this random phenonmenon.

I) What is the random event?

II) State the Assumptions:• 1)

• 2)

What will the random variable X represent:In words:

Values: X = { }

• Assign probabilities to the values of X.• X P(X)• 2 1/36• 3• 4• 5• 6• 7• 8• 9• 10• 11• 12

To answer the question (What is the probability of winning at craps on the first roll, we need to findP(x = 7 or x= 11)Are these disjoint events?disjoint events so P(x=7 or x=11)=

What did our relative frequency model predict from the activity?

Do we expect this type of variation?

Construct the probability model for this game.

X P(X)$3 1/8$4 1/14 Prizes = 63% of sales$5 1/22$6 1/28$8 1/76$10 1/76$25 1/100$50 1/280$100 1/4000$1000 1/240000$25000 1/480000

Is this a good probability model? If not what is missing?

Is this a discrete probability model? Why?

a) Express “ the unit has five or more rooms in terms of X What is the probability of this event?

b) Express the event X > 5 in words. What is P(x>5) ?

c) What important fact about discrete random variables does comparing your answers to a and b above illustrate?

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