6.1Discrete Random Variables

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 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 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

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

● The number of pages in statistics textbooks 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 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

A value in an interval between 3 and 8 feet● The length of a country and western song

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● 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● 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,

thenP(1) = 1/6

Probability Distribution Properties of P(x) Since P(x) form a probability

distribution, they must satisfy the rules of probability 0 ≤ P(x) ≤ 1 Σ P(x) = 1

In 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

● 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 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)

A 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● The mean of a discrete random

variable isμX = 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) ]

● 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● 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 meanσX

2 = 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 isσX

2 = [ 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

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 .197

3 .320

4 .374

a) Verify that this is a discrete probability distribution

b) Draw a probability histogram

X P(x)

0 .035

1 .074

2 .197

3 .320

4 .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.

A life insurance company sells a $250,000 1-year term life insurance policy to a 20-year old female for $200. According to the National Vital Statistics Report, 56(9), the probability that the female survives the year is .999544. compute and interpret the expected value of this policy to the insurance company.