advertisement

Random Variables

Definition

For a given sample sample space S of some experiment, a random variable (rv) is any rule that associates a number with each outcome in S . In mathematical language, a random variable is a function whose domain is the sample space and whose range is the set of real numbers.

We use uppercase letters, such as X and Y to, denote random variables and use lowercase letters, such as x and y , to denote some particular value of the corresponding random variable . For example, X ( s ) = x means that value x is associated with the oucome s by the rv X .

Random Variables

Examples:

1. Assume we toss a coin. Then S = { H, T } . We can define a rv

X by

X ( H ) = 1 and X ( T ) = 0

2. A techincian is going to check the quality of 10 prodcuts. For each product the outcome is either successful (S) or defective (D).

Then we can define a rv Y by

Y =

(

1 , successful

0 , defective

Definition

Any random variable whose only possible values are 0 abd 1 is called a Bernoulli random variable .

Random Variables

More examples:

3. (Example 3.3) We are investigating two gas stations. Each has six gas pumps. Consider the experiment in which the number of pumps in use at a particular time of day is determined for each of the stations.

Define rv ’s X , Y and U by

X = the total number of pumps in use at the two stations

Y = the difference between the number of pumps in use at station 1 and the number in use at station 2

U = the maximum of the numbers of pumps in use at the two stations

If this experiment is performed and s = (3 , 4) results, then

X ((3 , 4)) = 3 + 4 = 7 , so we say that the observed value of X was x = 7 . Similarly, the observed value of Y would be y = 3 − 4 = − 1 , and the observed value of U would be u = max (3 , 4) = 4 .

Random Variables

More examples:

4. Assume we toss a coin until we get a Head. Then the sample space would be S = { H, TH, TTH, TTTH, . . .

} If we define a rv

X by X

X = the number we totally tossed

Then X ( { H } ) = 1 , X ( { TH } ) = 2 , X ( { TTH } ) = 3 , . . . , and so on.

In this case, the random variable X can be any positive integer, which in all is infinite.

5. Assume we are going to measure the length of 100 desks.

Define the rv Y by

Y = the length of a particular desk

Y can also assume infinitly possible values.

Random Variables

Definition

A dicrete random variable is an rv whose possible values either constitute a finite set or else can be listed in an infinite sequence in which there is a first element, a second element, and so on

(“countably” infinite).

A random variable is continuous if both of the following apply:

1. Its set of possible values consists either of all numbers in a single interval on the number line (possibly infinite in extent, e.g.,

( −∞ , ∞ ) ) or all numbers in a disjoint union of such intervals

(e.g., [0 , 10] ∪ [20 , 30]).

2. No possible value of the variable has positive probability, that is,

P ( X = c ) = 0 for any possible value c.

Probability Distributions for Discrete RV

An example:

Assume we toss a coin 3 times and record the outcomes. Let X i be a random variable defined by

X i

=

(

1 , if the i th

0 , if the i th outcome is Head; outcome is Tail;

Let X be the random variable such that X = X

1

+ X

2

+ X

3

, then

X represents the total number of Heads we could get from the experiment.

If the probability for getting a Head for each toss is 0.7, then the probabilities for all the outcomes are tabulated as following: s x

HHH

3

HHT

2

HTH

2

HTT

1

THH

2

THT

1

TTH

1

TTT

0 p ( x ) 0.343

0.147

0.147

0.063

0.147

0.063

0.063

0.027

Probability Distributions for Discrete RV

Example continued: s x

HHH

3

HHT

2

HTH

2

HTT

1

THH

2

THT

1

TTH

1

TTT

0 p ( x ) 0.343

0.147

0.147

0.063

0.147

0.063

0.063

0.027

We can re-tabulate it only for the x values: x 0 1 2 3 p ( x ) 0.027

0.189

0.441

0.343

Now we can answer various questions.

The probability that there are at most 2 Heads is

P ( X ≤ 2) = P ( x = 0 or 1 or 2) = p (0) + p (1) + p (2) = 0 .

657

The probability that the number of Heads are is strictly between 1 and 3 is

P (1 < X < 3) = P ( X = 2) = p (2) = 0 .

441

Probability Distributions for Discrete RV

Definition

The probability distribution or probability mass function ( pmf ) of a discrete rv is defined for every number x by p ( x ) = P ( X = x ) = P (all s ∈ S : X ( s ) = x ) .

In words, for every possible value x of the random variable, the pmf specifies the probability of observing that value when the experiment is performed. (The conditions p ( x ) ≥ 0 and

P all possible x p ( x ) = 1 are required for any pmf .)

Probability Distributions for Discrete RV

Example 3.8

Six lots of components are ready to be shipped by a certain supplier. The number of defective components in each lot is as follows:

Lot 1 2 3 4 5 6

Number of defectives 0 2 0 1 2 0

One of these lots is to be randomly selected for shipment to a particular customer. Let X be the number of defectives in the selected lot.

The three possible X values are 0, 1 and 2. The pmf for X is

3 p (0) = P ( X = 0) = P ( lot 1 or 3 or 6 is selected ) = = 0 .

500

6 p (1) = P ( X = 1) = P ( lot 4 is selected ) =

1

= 0 .

167

6 p (2) = P ( X = 2) = P ( lot 2 or 5 is selected ) =

2

6

= 0 .

333

Probability Distributions for Discrete RV

Example 3.10:

Consider a group of five potential blood donors — a, b, c, d, and e

— of whom only a and b have type O+ blood. Five blood smaples, one from each individual, will be typed in random order until an O+ individual is identified. Let the rv Y = the number of typings necessary to identify an O+ individual. Then what is the pmf of Y ?