Random Variables A

random variable is a numeric measure of the outcome of a probability experiment

Random variables reflect measurements that can change as the experiment is repeated Random variables are denoted with capital letters, typically using X (and Y and Z …) Values are usually written with lower case letters, typically using x (and y and z ...)

Examples (Random Variables) ●

Tossing four coins and counting the number of heads

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The number could be 0, 1, 2, 3, or 4 The number could change when we toss another four coins

Measuring the heights of students

The heights could change from student to student

Discrete Random Variable ●

A discrete random variable is a random variable that has either a finite or a countable number of values

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A finite number of values such as {0, 1, 2, 3, and 4} A countable number of values such as {1, 2, 3, …}

Discrete random variables are designed to model discrete variables (see section 1.2) Discrete random variables are often “counts of …”

Example (Discrete Random Variable) ●

The number of heads in tossing 3 coins (a finite number of possible values)

There are four possible values – 0 heads, 1 head, 2 heads, and 3 heads A finite number of possible values – a discrete random variable This fits our general concept that discrete random variables are often “counts of …”

Discrete Random Variables ● ●

Other examples of discrete random variables The possible rolls when rolling a pair of dice

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The number of pages in statistics textbooks

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A finite number of possible pairs, ranging from (1,1) to (6,6) A countable number of possible values

The number of visitors to the White House in a day

A countable number of possible values

Continuous Random Variable ●

A continuous random variable is a random variable that has an infinite, and more than countable, number of values

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The values are any number in an interval

Continuous random variables are designed to model continuous variables (see section 1.1) Continuous random variables are often “measurements of …”

Example (Continuous Random Variable) ● ●

An example of a continuous random variable The possible temperature in Chicago at noon tomorrow, measured in degrees Fahrenheit

The possible values (assuming that we can measure temperature to great accuracy) are in an interval The interval may be something like (–20,110) This fits our general concept that continuous random variables are often “measurements of …”

Continuous Random Variables ● ●

Other examples of continuous random variables The height of a college student

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The length of a country and western song

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A value in an interval between 3 and 8 feet

A value in an interval between 1 and 15 minutes

The number of bytes of storage used on a 80 GB (80 billion bytes) hard drive

Although this is discrete, it is more reasonable to model it as a continuous random variable between 0 and 80 GB

Probability Distribution ●

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The probability distribution of a discrete random variable X relates the values of X with their corresponding probabilities A distribution could be

In the form of a table In the form of a graph In the form of a mathematical formula

Probability Distribution ●

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If X is a discrete random variable and x is a possible value for X, then we write P(x) as the probability that X is equal to x Examples

In tossing one coin, if X is the number of heads, then P(0) = 0.5 and P(1) = 0.5 In rolling one die, if X is the number rolled, then P(1) = 1/6

Probability Distribution Properties

of P(x) Since P(x) form a probability distribution, they must satisfy the rules of probability

In

0 ≤ P(x) ≤ 1 Σ P(x) = 1

the second rule, the Σ sign means to add up the P(x)’s for all the possible x’s

Probability Distribution

An example of a discrete probability distribution

All of the P(x) values are positive and they add up to 1

NOT a Probability Distribution ●

An example that is not a probability distribution

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Two things are wrong

P(5) is negative The P(x)’s do not add up to 1

Probability Histogram A

probability histogram is a histogram where

A

The horizontal axis corresponds to the possible values of X (i.e. the x’s) The vertical axis corresponds to the probabilities for those values (i.e. the P(x)’s)

probability histogram is very similar to a relative frequency histogram

Probability Histogram An

example of a probability histogram

The

histogram is drawn so that the height of the bar is the probability of that value

Mean of a Probability Distribution ●

The mean of a probability distribution can be thought of in this way:

There are various possible values of a discrete random variable The values that have the higher probabilities are the ones that occur more often The values that occur more often should have a larger role in calculating the mean The mean is the weighted average of the values, weighted by the probabilities

Mean of a Discrete Random Variable ●

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The mean of a discrete random variable is μX = Σ [ x • P(x) ] In this formula

x are the possible values of X P(x) is the probability that x occurs Σ means to add up these terms for all the possible values x

Mean ●

Example of a calculation for the mean

[ x • P(x) ]

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Add: 0.2 + 1.2 + 0.5 + 0.6 = 2.5 The mean of this discrete random variable is 2.5

Law of Large Numbers ●

The mean can also be thought of this way (as in the Law of Large Numbers)

If we repeat the experiment many times If we record the result each time If we calculate the mean of the results (this is just a mean of a group of numbers) Then this mean of the results gets closer and closer to the mean of the random variable

Expected Value ● ●

The expected value of a random variable is another term for its mean The term “expected value” illustrates the long term nature of the experiments – as we perform more and more experiments, the mean of the results of those experiments gets closer to the “expected value” of the random variable

Variance ● ●

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The variance of a discrete random variable is computed similarly as for the mean The mean is the weighted sum of the values μX = Σ [ x • P(x) ] The variance is the weighted sum of the squared differences from the mean σX2 = Σ [ (x – μX)2 • P(x) ] The standard deviation, as we’ve seen before, is the square root of the variance … σX = √ σX2

Variance The

variance formula σX2 = Σ [ (x – μX)2 • P(x) ] can involve calculations with many decimals or fractions An equivalent formula is σX2 = [ Σ x2 • P(x) ] – μX2 This formula is often easier to compute

Good News! The

variance can be calculated by hand, but the calculation is very tedious Whenever possible, use technology (calculators, software programs, etc.) to calculate variances and standard deviations See Handout

Summary Discrete

random variables are measures of outcomes that have discrete values Discrete random variables are specified by their probability distributions The mean of a discrete random variable can be interpreted as the long term average of repeated independent experiments The variance of a discrete random variable measures its dispersion from its mean

Let’s Try Some… Page Be

300 – 303

Calculator Ready!

Determine whether the random variable is discrete or continuous. State the possible values of the random variable. a)

The amount of rain in Seattle during April.

b)

The number of fish caught during a fishing tournament

c)

The number of customers arriving at a bank between noon and 1pm

d)

The time required to download a file from the internet

Determine whether the distribution is a discrete probability distribution. X

P(x)

100

.1

200

.25

300

.2

400

.3

500

.1

In the following probability distribution, the random variable X represents the number of activities a parent of a K-5th grade student is involved in

X P(x) 0 .035 1 .074 2 3 4

.197 .320 .374

a) Verify that this is a discrete probability distribution b) Draw a probability histogram

X P(x) 0 .035 1 .074 2 3 4

.197 .320 .374

C) Compute and interpret the mean of the random variable X.

D) Compute the variance of random variable X.

E) Compute the standard deviation of random variable x. F) What is the probability that a randomly selected student has a parent involved in 3 activities. G) What is the probability that a randomly selected student has a parent involved in 3 or 4 activities.

An investment counselor calls with a hot stock tip. He believes that if the economy remains strong, the investment will result in a profit of $50,000. If the economy grows at a moderated pace, the investment will result in a profit of $10,000. however, if the economy goes into recession, the investment will result in a loss of $50,000. You contact an economist who believes that there is a 20% probability the economy will remain strong, a 70% probability that the economy will grow at a moderate pace, and a 10% probability that the economy will slip into recession. What is the expected profit from this investment.