Chapter 6

Random Variables

Make a Sample Space for Tossing a Fair Coin 3 times.

How many “heads?”

O heads

1 head

2 heads

3 heads

Probability Distribution Table# of Heads (Value)

0 1 2 3

Probability

1. What is the probability of getting AT LEAST one head?

2. What is the probability of getting less than 3 heads?

3. What is the probability of getting one or less?

HistogramDraw a Histogram that represents the probability distribution.

Random Variable

• A number that describes the outcomes of a chance process.

• In the coin toss example, the random variable was the # of heads. (0,1,2 or 3)

2 types• Discrete-• Continuous-• Tossing a coin- continuous or discrete?

Discrete Random Variables (DRV)

• The probability distribution of a DRV must follow these rules:– Every probability must be a number between __

and __(fractions and decimals are ok)– The sum of the probabilities is __.– To find the value of an event, ____ the

probabilities of the values that make up the event.

CYU page 344

NC State posts the grade distributions for its courses online. Students taking Stats 101 last semester received 26% As, 42% Bs, 20% Cs, 10% Ds, and 2 % Fs. The student’s grade on a 4 point scale is a discrete random variable X with this probability distribution.

Value of x 0 1 2 3 4

Probability 0.02 0.10 0.20 0.42 0.26

QuestionsValue of x 0 1 2 3 4

Probability 0.02 0.10 0.20 0.42 0.26

1. Say in words what the meaning of P(x > 3) is. What is this probability?

2. Write the event “the student got a grade worse than C” in terms of values of the random variable X. What is the probability of this event?

3. Sketch a graph and describe what you see.

Mean of a DRV

The mean of a DRV is the expected value.

In other words, if you tossed a coin 3 times, what would you expect to happen?

If you took Stats 101 at NC State, what would you expect to happen?

Let’s use the example of a roulette wheel.

Roulette

• The rules:– 38 numbered slots. (1-36, 0 and 00)– Half red, half black– 0 and 00 are green– You can bet on a color, number, even/odd, etc.

Expected Value

• Suppose you bet $1 on red. If the ball lands in a red slot, you will get the dollar back, plus an additional dollar for winning the bet.

• If the ball lands in a different colored slot, you will lose the dollar.

• Define the random variable:• Possible values of x:• Probabilities of each possible value:

Probability Distribution

The “expected value” or mean is the player’s average gain.

NHL goalsGoals 0 1 2 3 4

Probability 0.061 0.154 0.228 0.229 0.173

5 6 7 8 9

0.094 0.041 0.015 0.004 0.001

Find the mean (expected value).

Try it on the calculator.

• See page 348 for instructions.

Standard Deviation of a DRV

• Find the variance and take the square root.• To find the variance, find the “average

squared deviation from the mean.”• In other words, find the deviation of each

value from the mean. Then, square each deviation. Finally, average them.

CYU page 349Cars Sold 0 1 2 3

Probability 0.3 0.4 0.2 0.1

Calculate and interpret the mean.

Calculate and interpret the standard deviation.

Continuous Random Variables (CRV)

• NOT possible to list all possible outcomes. • Examples: heights, weights, blood pressure,

etc.• Why is height a continuous variable?• Would shoe size be discrete or continuous or

discrete? Why?