slides – Math 132B

Definition of a Random Variable

A random variable is a mathematical model of a random quantity (measurement, count)

It is a function that assigns each outcome from a sample space a number.

Example: The number of dots on the side of a die that is facing up.

Example: The total number of dots on the sides of two dice that are facing up.

Example: The number of infants that choose the friendly character.

Example: The height of a plant randomly selected from a field.

Example

the number of heads when flipping 3 coins:

Example

models “the number of heads in 3 tosses of a fair coin”.

can take on the values 0, 1, 2, 3.

3 coin tosses

More examples:

Taking a token 3 times from a bag with one red and one blue token, with replacement. Counting the number of red tokens.
Asking 3 people to randomly select one of two characters, when they have no preference. Counting the number of times the friendly character was selected.
Randomly selecting 3 people from a large crowd that has 50% males and 50% females. Counting the number of females.
Randomly selecting 3 plants from a field in which 50% of the plants have some specific genetic mutation. Counting the number of plants with the mutation.

Mathematically, all of those are the same.

Distribution of a Random Variable

The distribution of a discrete random variable is the collection of its values and the probabilities associated with those values.

The probability distribution for is as follows:

	0	1	2	3
	1/8	3/8	3/8	1/8

The probabilities must add up to 1

Bar graph showing a distribution

	0	1	2	3
	1/8	3/8	3/8	1/8

Using a simulation to approximate a probability distribution

tosses <- do (10000) * rflip(3)
gf_props(~heads, data=tosses)

Expectation of a Random Variable

If has outcomes , …, with probabilities , …, , the expected value of , also called the mean of , is the sum of each outcome multiplied by its corresponding probability:

The Greek letter may be used in place of the notation and is sometimes written .

Expectation…

In the coin tossing example,

	0	1	2	3

Expectation…

Intuitively, the expected value of is approximately the number you would get if you took a lot of values of and calculated the mean.

tosses <- do (10000) * rflip(3)
mean(~heads, data=tosses)

[1] 1.4923

Variance and SD of a Random Variable

If takes on outcomes , …, with probabilities , …, and has the expected value , then the variance of , denoted by or , is

The standard deviation of , written as or , is the square root of the variance. It is sometimes written .

Variance and SD…

In the coin tossing example,

	0	1	2	3
	1/8	3/8	3/8	1/8

The standard deviation is .

Standard Deviation …

Again, the standard deviation of is approximately the number you would get if you took a lot of values of and calculated the standard deviation of the data.

tosses <- do (10000) * rflip(3)
sd(~heads, data=tosses)

[1] 0.8643192

Another example

Suppose your roll three fair six sided dice. Let .

What are the possible values of ?
What is the probability distribution for ?
What is the expected value of ?
What is the variance of ?

Common theme

Number of heads when tossing three fair coins
Number of sixes when rolling three fair dice
Number of heads when tossing 16 fair coins
Number of infants choosing the helper, provided infants choose randomly.

All these can be modeled by so-called binomial random variables.

Binomial Random Variables

is a binomial random variable if it represents the number of successes in replications of an experiment where

Each replicate is independent of the other replicates.
Each replicate has two possible outcomes: either success or failure.
The probability of success in each replicate is constant.

This is also called a binomial process.

A binomial random variable takes on values .

The numbers and are called the parameters of the distribution.

We write .

The Binomial Distribution

Suppose . What is ?

independent repetitions
successes, each with probability
failures, each with probability

But also

or it may be that the first is success, then there are two failures, then a success, and so on.

How many different ways can we choose successes out of repetitions?

The Binomial Coefficient

The binomial coefficient is the number of ways to choose items from a set of size , where the order of the choices is ignored.

Mathematically,

Examples

Calculate
Calculate
Calculate
Calculate

Formula for the binomial distribution

Let = number of successes in trials, and .

Parameters of the distribution:

= number of trials
= probability of success

Examples

Let and .

Calculate .
Let and .

Calculate .
Let and .

Calculate .

(Hint: )

Mean and SD for a binomial random variable

For a binomial distribution with parameters and , it can be shown that:

Mean =
Standard Deviation =

The derivation is not shown here nor in the text; it will not be asked for on a problem set or exam.

Binomial Probabilities in `R`

The function dbinom() is used to calculate .

dbinom(k, size=n, prob=p):

For example, if , the is

dbinom(2, size=3, prob=1/6)

[1] 0.06944444

The d in dbinom stands for distribution or density.

Binomial Probabilities in `R` …

The function pbinom() is used to calculate or .

:

pbinom(k, size=n, prob=p)
:

pbinom(k, size=n, prob=p, lower.tail = FALSE)

The p stands for probability.

`pbinom` examples:

if , then is

pbinom(13, size=16, prob=1/2)

[1] 0.9979095

while is

pbinom(13, size=16, prob=1/2, lower.tail=FALSE)

[1] 0.002090454

or, equivalently:

1 - pbinom(13, size=16, prob=1/2)

[1] 0.002090454