1-Random-Variables-and-Probability-Distributions.pdf

statistics and probability

1. EXPLORING RANDOM
VARIABLES AND PROBABILITY
DISTRIBUTIONS

2. DEFINITION
•A random variable is a function that associates a real number to each element in the
sample space. It is a variable whose values are determined by chance.
•A random variable is a numerical quantity that is derived from the outcomes of random
experiments.
•A random variable is called discrete if it has either a finite or a countable number of
possible values. A random variable is called continuous if its possible values contain a
whole interval of numbers. (Malate, 2018)

3. CLASSIFY THE FOLLOWING RANDOM VARIABLES AS
DISCRETEOR CONTINUOUS
1.The number of defective computers produced by a manufacturer
2.The weight of newborns each year in a hospital
3.The number of siblings in a family or region
4.The amount of paint utilized in abuilding project
5.The number of dropout in a school district
6.The speed of a car
7.The number of female athletes
8.The time needed to finish the test
9.The amount of sugar in a cup of coffee

4. FIND THE PROBABILITYOF THE FOLLOWING EVENTS
1. Getting an even number in a single roll of a die
2. Getting a sum of 6 when two dice are rolled
3. Getting an ace when a card is drawn from a deck
4. The probability that all children are boys if a couple has three children
5. Getting an odd number and a tail when a die is rolled and a coin is tossed
simultaneously
6. Getting a sum of 11 when two dice are rolled
7. Getting a red queen when a card is drawn from a deck

5. NUMBER OF TAILS
STEPS:
1. Determine the sample space. Let H represent head and T represents tail.
2. Count the number of tails in each outcome in the sample space and assign this number to
this outcome.
3. Assign probability values P(Y) to each value of the random variable.
•Suppose three coins are tossed. Let Y be the random variable representing the
number of tails that occur. Find the probability of each of the values of the
random variable Y.

6. DEFINITION
•A discrete probability distribution or a probability mass function consists
of the values a random variable can assume and the corresponding
probabilities of the values.
Number of
Tails (Y)
0 1 2 3
Probability P(Y)1/8 3/8 3/8 1/8
•The Probability Distribution or the Probability Mass Function of Discrete
Random Variable Y

7. EXAMPLE 2
Two balls are drawn in succession from an urn
containing 5 red balls and 6 blue balls. Let Z be the
random variable representing the number of blue balls.
Construct the probability distribution of the random
variable Z.

8. HISTOGRAM (BAR GRAPH)

9. EXAMPLE 3
Suppose three cellphones are tested at random.
Let D represent the defective cellphone and let N
represent the non-defective cellphone. If we let X be
the random variable for the number of defective
cellphones, construct the probability distribution of
the random variable X.

10. PROPERTIES OF A PROBABILITY DISTRIBUTION
•The probability of each value of the random variable must be
between or equal to 0 and 1. In symbol, we write 0≤??????(??????)≤1.
•The sum of the probabilities of all values of the random variable
must be equal to 1. In symbol, we write it as σ????????????=1.

11. DETERMINE WHETHER THE DISTRIBUTION
REPRESENTS A PROBABILITY DISTRIBUTION.
X 1 5 8 7 9
P(X)1/31/31/31/31/3

12. DETERMINE WHETHER THE DISTRIBUTION
REPRESENTS A PROBABILITY DISTRIBUTION.
X 0 2 4 6 8
P(X)1/61/61/31/61/6

13. DETERMINE WHETHER THE DISTRIBUTION
REPRESENTS A PROBABILITY DISTRIBUTION.
X 1 2 3 5
P(X)1/41/81/41/8

14. DETERMINE WHETHER THE DISTRIBUTION
REPRESENTS A PROBABILITY DISTRIBUTION.
X 1 3 5 7
P(X)0.350.250.220.12