This document discusses random variables and probability distributions. It defines a random variable as a rule that assigns numerical values to outcomes of an experiment. Random variables can be discrete or continuous. A probability distribution lists the probability values associated with each possible value in the range of a discrete random variable. Examples are provided to illustrate range spaces of random variables and probability distributions, including experiments with coin tosses, dice rolls, and basketball games. The properties of a discrete probability distribution are also mentioned.
2. RANDOM VARIABLES
• A random variable is a rule that assigns a numerical
value or characteristic to an outcome of an
experiment. It is essentially a variable, usually denoted
by any capital letter of the alphabet because its value
is not constant or assumes different values due to
chance.
• Generally, there are two categories of random
variables: discrete and continuous random variables.
• For a quick comparison, the values of a discrete
random variable are usually counts and those of a
continuous random variable are measurements.
3. Objectives:
After this lesson, you should be able to:
• Determine possible values of random
variable (review);
• Illustrate the properties of discrete
random variables; and
• Compute probabilities corresponding to a
discrete random variable.
4. RANGE SPACE
• The possible values of a random variable or the range space are values that are
obtained from functions that assigns a real number to each point of a sample
space.
• Examples:
Identify the range space of the following random variables:
1. Two fair coins are tossed simultaneously and X is defined as the number of
heads that appear.
S= {HH, HT, TH, TT}
X={0,1,2}
5. RANGE SPACE
• The possible values of a random variable or the range space are values that are
obtained from functions that assigns a real number to each point of a sample space.
• Examples:
Identify the range space of the following random variables:
2. A pair of dice is thrown together and Y is defined as the sum of two numbers that
appear.
S = {(1,1) (1,2) (1,3) (1,4) (1,5) (1,6) (2,1) (2,2) (2,3) (2,4) (2,5) (2,6)
(3,1) (3.2) (3,3) (3,4) (3,5) (3,6) (4,1) (4,2) (4,3) (4,4 (4,5) (4,6)
(5,1) (5,2) (5,3) (5,4) (5,5) (5,6) (6,1) (6,2) (6,3) (6,4) (6,5) (6,6)}
Y = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
6. RANGE SPACE
• The possible values of a random variable or the range space are values that are
obtained from functions that assigns a real number to each point of a sample
space.
• Examples:
Identify the range space of the following random variables:
3. A basketball team plays five consecutive games and Z is defined as the number
of wins.
S ={LLLLL, WLLLL, LWLLL, LLWLL, LLLWL, LLLLW, WWLLL, WLWLL, WLLWL, WLLLW, LWWLL,
LWLWL, LWLLW, LLWWL, LLWLW, LLLWW, WWWLL, WWLWL, WWLLW, WLWWL,
WLLWW, WLWLW, LLWWW, LWLWW, WLLWW, WWWWL, LWWWW, WLWWW,
WWLWW, WWWLW, WWWWW}
Z= {0, 1, 2, 3, 4, 5}
7. PROBABILITY DISTRIBUTIONS
• A probability distributions, also known as probability mass function, is a
table that gives a list of probability values along with their associated
value in the range space of a discrete random variable.
• Illustrative Examples:
Experiment 1: Two fair coins are tossed simultaneously.
Sample Space: HH, HT, TH, TT
X: the number of heads that appear
X 0 1 2
P(X) 1/4 2/4 1/4
