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probability distributions
USING RANDOM VARIABLES TO SIMPLIFY STATEMENTS
TAKE We can use a variable to represent a long statement so that we no longer
NOTE! have to write it again and again. In probability, this variable representing a
statement that can assign a number to each outcome of an experiment is
called a RANDOM VARIABLE (denoted by capital letters X, Y, Z, etc.)
EXAMPLE 1. Suppose we roll a die.
For EXAMPLE 1: What are the outcomes?
1, 2, 3, 4, 5, 6
Make a statement about an outcome that will give a number:
Since the outcomes are already single numbers, we can simply choose the statement:
“The outcome of rolling our die” or “The result of rolling a die”
Define the random variable X, as:
X = The outcome of rolling our die
Using this, we can make various other probability statements regarding our experiment:
P(X = 5)
The probability that the outcome of rolling our die is 5
PROBABILITY DISTRIBUTIONS — USING RANDOM VARIABLES TO SIMPLIFY STATEMENTS
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Define the random variable X, as:
X = The outcome of rolling our die
P(X < 3)
The probability that the outcome of rolling our die is less than 3
P(X > 7)
The probability that the outcome of rolling our die is greater than 7
P(2 < X ≤ 5)
The probability that the outcome of rolling our die is between 2 and 5,
(excluding 2 and including 5)
EXAMPLE 2. Suppose we toss a coin five times. Define a random variable for this experiment.
Outcomes?
Possible random variables:
HTHTH, THTTH, HHHTH, TTTTT, HHHH, etc. etc. etc.
X = The number of heads that turned up
Y = The number of tails that turned up
EXAMPLE 3. Suppose we randomly select ten cards from a standard deck of playing cards.
Define a random variable for this experiment.
Outcomes?
Possible random variables:
(10 different cards), (10 different cards), etc. etc. etc.
X = The number of red cards included in the selection
Y = The number of hero cards included in the selection
PROBABILITY DISTRIBUTIONS — USING RANDOM VARIABLES TO SIMPLIFY STATEMENTS
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NOTE!
The frequency table of a certain variable can be regarded as the outcomes of
the experiment of randomly selecting an individual element of the population.
EXAMPLE 4. The ages of all 2500 teen-age residents of Brgy. Recto are recorded as a part of a
local study on juvenile delinquency. Define the random variable for the experiment
of randomly selecting a teen-age resident of Brgy. Recto.
Age
13
14
15
16
17
18
19
Frequency
336
364
426
410
507
334
123
Outcomes?
Possible
Random Variables:
Residents of Brgy. Recto of age 13 up to 19
X = Age of a randomly selected teen-ager of Brgy. Recto
variable
measured?
NOTE!
“of a randomly
selected”
one member of
the population
For probability experiments using actual survey results, the
random variable can be easily defined using the pattern above.
P(X<15) = Probability that the age of a randomly
selected teen-ager is less than 15
= 700/2500
P(14<X<18) = Probability that the age of a randomly = 1343/2500
selected teen-ager is from 15 to 17
PROBABILITY DISTRIBUTIONS — USING RANDOM VARIABLES TO SIMPLIFY STATEMENTS
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HOW IS THE RANDOM VARIABLE RELATED TO THE VARIABLE?
TAKE A variable has a frequency distribution. Again, this is a table or a histogram
NOTE! showing the frequency pattern of the data when it is arranged into groups.
EXAMPLE. The ages of all 2500 teen-age residents of Brgy. Recto are obtained as a part of a
local study on juvenile delinquency.
Frequency
600
13
336
500
14
364
15
426
16
410
17
507
18
334
19
123
Frequency
Age
400
300
200
100
0
13
14
15
16
PROBABILITY DISTRIBUTIONS — HOW IS THE RANDOM VARIABLE RELATED TO THE VARIABLE?
17
18
19
Age
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TAKE The frequency distribution of a variable can be converted into a relative
NOTE! frequency distribution (where, the frequencies are simply converted to %).
This relative frequency distribution is also called PROBABILITY DISTRIBUTION.
First, we define the random variable:
X
Age
P(X=?)
Frequency
X = Age of a randomly selected teen-age resident
of Brgy. Recto
P(X)
0.25
13
P(X=13)
0.1344
336 = 336/2500
14
P(X=14)
0.1456
364 = 364/2500
15
P(X=15)
0.1704
426 = 426/2500
0.15
16
P(X=16)
0.1640
410 = 410/2500
0.1
17
P(X=17)
0.2028
507 = 507/2500
18
334 = 334/2500
P(X=18)
0.1336
19
P(X=19)
0.0492
123 = 123/2500
0.2
0.05
X
0
13
14
15
16
17
18
19
TAKE The frequency histogram of a variable is congruent to its corresponding
NOTE! probability distribution. Also, the sum total of all probabilities is 1.
PROBABILITY DISTRIBUTIONS — HOW IS THE RANDOM VARIABLE RELATED TO THE VARIABLE?
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EXAMPLES OF PROBABILITY DISTRIBUTIONS
EXAMPLE. A coin is tossed four times.
Define the random variable:
Sample space:
X = No. of ‘heads’ that turned up
HHHH HHHT HHTH HTHH THHH
HHTT
HTTH
TTHH
THTH
TTHT
TTTH
TTTT
THHT
HTHT
PROBABILITY TABLE
HTTT
THTT
PROBABILITY HISTOGRAM
P(X)
0.4
X
P(X=?)
0
0.0625
1/16
0.3
1
4/16
0.25
0.25
2
0.375
6/16
0.15
3
4/16
0.25
0.1
4
0.0625
1/16
0.35
0.2
0.05
0
0
1
PROBABILITY DISTRIBUTIONS — EXAMPLES OF PROBABILITY DISTRIBUTIONS
2
3
4
X
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As a part of a medical study, the heights
(in cms) of 100 adult male individuals in
Brgy. Recto are measured.
Freq
153.5–156.5
8
156.5–159.5
18
159.5–162.5
24
162.5–165.5
28
165.5–168.5
22
PROBABILITY HISTOGRAM
P(X)
0.3
X
P(X=?)
153.5–156.5
0.08
156.5–159.5
0.18
159.5–162.5
0.24
0.1
162.5–165.5
0.28
0.05
165.5–168.5
0.22
0
0.25
0.2
0.15
PROBABILITY DISTRIBUTIONS — EXAMPLES OF PROBABILITY DISTRIBUTIONS
168.5
165.5
162.5
X
159.5
PROBABILITY TABLE
Height (cms)
X = Height of a randomly selected adult male
156.5
Define the random variable:
153.5
EXAMPLE.
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SOME IMPORTANT PROBABILITY DISTRIBUTIONS
TAKE There are so many real-world situations whose probability distributions can
NOTE! be satisfactorily modeled by certain probability formulas!
THE UNIFORM PROBABILITY DISTRIBUTION
EXAMPLE 1.
Consider the experiment of randomly picking a ball from a box containing 4 balls
labeled 1, 2, 3 and 4.
X = Label of the ball picked
Define the random variable:
X
P(X=?)
1
1/4
2
1/4
3
1/4
4
1/4
P(X)
0.3
0.25
0.2
0.15
0.1
0.05
0
1
2
3
SOME IMPORTANT PROBABILITY DISTRIBUTIONS — THE UNIFORM PROBABILITY DISTRIBUTION
4
X
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EXAMPLE 2.
Consider the experiment of randomly picking a ball from a box containing 6 balls
labeled 1, 2, 3, 4, 5 and 6.
X = Label of the ball picked
Define the random variable:
P(X)
0.18
X
P(X=?)
1
1/6
2
1/6
0.12
3
1/6
0.1
4
1/6
5
1/6
0.04
6
1/6
0.02
0.16
0.14
0.08
0.06
0
X
1
2
3
4
5
6
TAKE Generalizing this example, we see that, if we have a box containing k balls
NOTE! (labelled 1, 2, 3, …, k), then the each ball has an equal chance of getting
picked as any other ball, and the probability is 1/k.
SOME IMPORTANT PROBABILITY DISTRIBUTIONS — THE UNIFORM PROBABILITY DISTRIBUTION
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TAKE
NOTE!
THE BASIC PATTERN OF THE UNIFORMLY DISTRIBUTED PROBABILITY
If you can liken a process to simply randomly picking one item from
a box containing N such items labelled 1, 2, 3, …, N, then this process is
said to have a UNIFORM PROBABILITY DISTRIBUTION.
Here, the random variable is defined as:
X = The label of the item picked
and the probability formula is:
P(X  k)  1
k
EXAMPLE 3.
where k is any label from 1 to N.
Consider the experiment of rolling a die.
Think!: Is rolling a die similar to randomly picking one item
from a box with N=6 items labelled 1, 2, 3, 4, 5, 6?
Define the random variable:
YES!
X = Result (label) of tossing the die
P(X = 1) = 1/6
P(X = 3) = 1/6
P(X = 5) = 1/6
P(X = 2) = 1/6
P(X = 4) = 1/6
P(X = 6) = 1/6
SOME IMPORTANT PROBABILITY DISTRIBUTIONS — THE UNIFORM PROBABILITY DISTRIBUTION
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THE BINOMIAL PROBABILITY DISTRIBUTION
EXAMPLE 3.
Consider the experiment of tossing a coin four (4) times, and suppose we are
interested in the number of heads that turned up.
Define the random variable:
Sample space:
X = The number of Heads that turned up
HHHH HHHT HHTH HTHH THHH
HHTT
HTTH
TTHH
THTH
TTHT
TTTH
TTTT
THHT
HTHT
HTTT
THTT
What is P(X=3)? 4/16 or 0.25
X=3:
HHHT, HHTH, HTHH, THHH (4)
What is P(X=2)?
X=2:
6/16 or 0.375
HHTT, HTTH, TTHH, THTH, THHT, HTHT
(6)
What is P(X=4)? 1/16 or 0.0625
X=3:
HHHH
(1)
SOME IMPORTANT PROBABILITY DISTRIBUTIONS — THE BINOMIAL PROBABILITY DISTRIBUTION
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TAKE
NOTE!
THE BASIC PATTERN OF THE BINOMIAL PROBABILITY DISTRIBUTION
If a process has exactly two outcomes: a desired outcome and an
unwanted one, and is repeated a definite number of times, yet still these
two outcomes remain unaffected through all these repetitions. Then this
repeated process is said to have a BINOMIAL PROBABILITY DISTRIBUTION.
Here, the random variable is defined as:
X = The number of desired outcomes after repeating the
process N times
and the probability formula is:
 N
P(X  k)     pk  (1  p)N  k
k
where k is the number of desired outcomes
and p is the probability of one desired outcome of the process.
SOME IMPORTANT PROBABILITY DISTRIBUTIONS — THE BINOMIAL PROBABILITY DISTRIBUTION
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EXAMPLE 3.
Consider the experiment of tossing a coin four (4) times, and suppose we are
interested in the number of heads that turned up.
Think!: Is this experiment a repeated process?
Does this process have exactly two outcomes, and we
are interested in one of these outcomes?
Are these two outcomes unaffected by the repetitions?
Define the random variable:
X = The number of Heads that turned up
after tossing the coin four times (N=4)
A. What is the probability of getting 2 heads after tossing the coin four times?
 N 2
N 2
P(X=2) =    p  (1  p)
 2
p = probability of one desired outcome (one head)
= 1/2 or 0.5
4
=    (0.5)2  (1  0.5)4  2
 2
= 0.375
SOME IMPORTANT PROBABILITY DISTRIBUTIONS — THE BINOMIAL PROBABILITY DISTRIBUTION
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B. What is the probability of getting 3 heads after tossing the coin four times?
4
P(X=3) =    (0.5)3  (1  0.5)4  3 = 0.25
 3
C. What is the probability of getting at most 2 heads after tossing the coin four times?
Remember!: Getting at most 2 heads means: getting no heads (0), one head, or two heads.
Getting at least 2 heads means: getting two heads, three heads, or four heads.
4
P(X=2) =    (0.5)2  (1  0.5)4  2 = 0.375
 2
4
P(X=1) =    (0.5)1  (1  0.5)4  1
 1
= 0.25
4
P(X=0) =    (0.5)0  (1  0.5)4  0 = 0.0625
0
(adding up) = 0.6875
SOME IMPORTANT PROBABILITY DISTRIBUTIONS — THE BINOMIAL PROBABILITY DISTRIBUTION
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EXAMPLE 4. Your professor gave a surprise multiple-choice quiz: 10 items, 4 choices per item.
You were unprepared, and decided to answer the questions by pure guessing.
After the quiz, you are anxious about your score.
A. Verify if this experiment has a binomial probability distribution. Define the random variable.
B. What is the probability of having guessed 3 items correctly?
C. What is the probability of having guessed all items incorrectly?
D. What is the probability of having guessed at least 8 items correctly?
EXAMPLE 5. The probability that a patient recovers from a delicate surgery is 0.86. Now, 7
patients are lined up for this operation. You are wondering about how many of
these (seven) patients will recover from this operation.
A. Verify if this experiment has a binomial probability distribution. Define the random variable.
B. What is the probability that at least 3 of these patients will recover?
C. What is the probability that 2 to 5 of these patients will recover?
SOME IMPORTANT PROBABILITY DISTRIBUTIONS — THE BINOMIAL PROBABILITY DISTRIBUTION
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