UNIT 6 – CHAPTER 66.1 – Discrete and Continuous Random Variables

DO YOU REMEMBER?

What is a probability distribution?

EXAMPLE: FLIPPING OUT!

Below is a probability distribution for flipping a coin four times and counting how many heads we see:

The number of heads we see is our Random Variable. In this problem we called it “X.”

Remember that a distribution shows what is possible and how often.

X 0 1 2 3 4

P(X) 1/16 4/16 6/16 4/16 1/16

DEFINITIONS

A random variable takes numerical values that describes the outcomes of some chance process.

The probability distribution of a random variable gives its possible values and their probabilities.

X 0 1 2 3 4

P(X) 1/16 4/16 6/16 4/16 1/16

DISCRETE RANDOM VARIABLE

A discrete random variable X takes a fixed set of possible values with gaps between.

For example: grades on the AP test (1, 2, 3, 4 ,5)

DICE, DICE EVERYWHERE!

Come grab a die from me.

Roll the die 20+ times, recording your results as you go

Find the average and standard deviation for your die rolls

Record your mean and standard deviation

Share your mean and standard deivation them on the board

EXPECTED VALUE (MEAN)

The mean of a discrete random variable X is also an average of the possible values of X.

To find the mean (expected value) of X, multiply each possible value by its probability then add all of the products.

Note:

EXAMPLE: A BLAST FROM THE PAST

Below is the score distribution for AP tests waaaaay back in 1997:

So we “expect” to see a score of 3.022. Notice that expected value should NOT be

rounded. Remember, this is an average!

X 1 2 3 4 5

P(X) .156 .221 .245 .196 .181

PRESS YOUR LUCK:

On an American roulette wheel there are 38 slots numbered 1 through 38. Slots are numbered 1-36 (half red half black) and two extra slots are labeled 0 or 00 and are green. If a player places a $1 bet on “red” the probability distribution for their net gain is as follows:

Find the expected value for net gain.

Value -$1 $1

Probability 20/38 18/38

STANDARD DEVIATION (AND VARIANCE) OF A DISCRETE RANDOM VARIABLE

When you use the mean (expected value) you will also use the standard deviation.

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Standard Deviation is the square root of the variance.

Note:

EXAMPLE: A BLAST FROM THE PAST… AGAIN

Below is the score distribution for AP tests waaaaay back in 1997… for a second time:X 1 2 3 4 5

P(X) .156 .221 .245 .196 .181

PRESS YOUR LUCK… AGAIN:

Using the same roulette example, find the standard deviation for net gain, when betting $1 on red:

Value -$1 $1

Probability 20/38 18/38

CONTINUOUS RANDOM VARIABLE

A continuous random variable, X, takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event.

For example – heights of all people in this room

YOU DECIDE! DISCRETE OR CONTINUOUS?

Time it takes you to clean your room

How much it costs for dinner

Inches of rainfall last month

Your birthday

The outcome of the roll of a dice

HOMEWORK

P. 353:2-8 even, 9, 12-18 even, 21-30