Random Variable - Advanced Engineering Math - Lecture Slides, Slides of Engineering Mathematics

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Random variable

Histogram of midterm score

5 10 15 20 25 30 35 40 45 0

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Midterm score

Formal definition of RV

• A random variable is not a variable. It is a

function from the sample space to the real

numbers.

– There is nothing random in the function itself. The

randomness comes from the probability measure

assigned to the events of the sample space.

• A random variable is often written as X(), where

 is an element in the sample space.

The real line

X()

Example: Midterm score

Class of ENGG2012B

Student A 23

The “midterm score” is a random variable. It maps a student to the corresponding midterm score.

Student B

Student C 35

If we randomly pick a student out of the total of 90 students, the probability that “midterm score = 23” is 2/90.

Notations

• The probability function Pr is sometime

regarded as a operator, which takes an event

as input and outputs a real number.

• If we want to emphasize that “Pr” is a function

with events as input, we shall write

for the probability of X() = 1, or simply

Probability distribution function

• Sometimes, we underlying sample space  is very complicated or

very large. We only care about the distribution of the random

variable, but not sample space .

• We may try to forget the sample space and work with the

probability mass function (pmf),

f ( i ) = Pr(X() = i ).

• If we want to emphasize that it is the pmf of random variable X(),

we can write fX ( i ).

• The pmf is sometimes called the “distribution of X”. However, in

probability, the term “distribution” usually refers to something else,

namely, F (z) = Pr(X()  z).

• In the remainder of this lecture, we assume that the random

variables take values in the integers, i.e., i is an integer.

Example: number of heads

• Toss 5 fair coins. Let Y() be the number of

heads. The pmf of Y() in tabular form is

i 0 1 2 3 4 5 f Y( i ) 1/32^ 5/32^ 10/32^ 10/32^ 5/32^ 1/