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.
Examples

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 ?