Random Variables

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Random Variables

• Budhi Setiawan
• Teknik Sipil - UNSRI

Learn how to characterize the pattern of the
distribution of values that a random variable may
have, and how to use the pattern to find
probabilities
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What is a Random Variable?

• Random Variable an outcome or event may be
  identified through the value(s) of a function,
  which usually denoted with a capital letter

If the value of X represent flood above mean
level, then X gt 7 meter stand for the occurrence
of floods above 7 meter

• Two different broad classes of random variables
• A continuous random variable can take any value
  in an interval or collection of intervals.
• A discrete random variable can take one of a
  countable list of distinct values.

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Example Random Variables at an Outdoor
Graduation or Wedding
Random factors that will determine how enjoyable
the event is Temperature continuous random
variable (any value, integer or decimal) Number
of airplanes that fly overhead discrete random
variable (integer only)
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Example Random VariablesProbability an Event
Occurs 3 Times in 3 Tries

• What is the probability that three tosses of a
  fair coin will result in three heads?
• Assuming boys and girls are equally likely, what
  is the probability that 3 births will result in 3
  girls?
• Assuming probability is 1/2 that a randomly
  selected individual will be taller than median
  height of a population, what is the probability
  that 3 randomly selected individuals will all be
  taller than the median?

Answer to all three questions 1/8.
Discrete Random Variable X number of times the
outcome of interest occurs in three independent
tries.
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Discrete Random Variables
X the random variable. k a number the discrete
r.v. could assume. P(X k) is the probability
that X equals k.
Discrete random variable can only result in a
countable set of possibilities often a finite
number of outcomes, but can be infinite.
Example Its Possible to Toss
Forever Repeatedly tossing a fair coin, and
define X number of tosses until the first
head occursAny number of flips is a possible
outcome. P(X k) (1/2)k
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Probability Distribution of a Discrete R.V.
Using the sample space to find probabilities
Step 1 List all simple events in sample
space. Step 2 Find probability for each simple
event. Step 3 List possible values for random
variable X and identify the value for each
simple event. Step 4 Find all simple events for
which X k, for each possible value k. Step 5
P(X k) is the sum of the probabilities for
all simple events for which X k.
Probability distribution function (pdf) X is a
table or rule that assigns probabilities to
possible values of X.
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ExampleHow Many Girls are Likely?
Family has 3 children. Probability of a girl is
?What are the probabilities of having 0, 1, 2,
or 3 girls?
Sample Space For each birth, write either B or
G. There are eight possible arrangements of B and
G for three births. These are the simple
events. Sample Space and Probabilities The eight
simple events are equally likely. Random Variable
X number of girls in three births. For each
simple event, the value of X is the number of Gs
listed.
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Example How Many Girls? (cont)
Value of X for each simple event
Probability distribution function for Number of
Girls X
Graph of the pdf of X
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Conditions for Probabilities for Discrete
Random Variables
Condition 1 The sum of the probabilities over
all possible values of a discrete random variable
must equal 1. Condition 2 The probability of
any specific outcome for a discrete random
variable must be between 0 and 1.
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Cumulative Distribution Function of a Discrete
Random Variable
Cumulative distribution function (cdf) for a
random variable X is a rule or table that
provides the probabilities P(X k) for any real
number k. Cumulative probability probability
that X is less than or equal to a particular
value.
Example Cumulative Distribution Function for
the Number of Girls (cont)
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Finding Probabilities for Complex Events
Example A Mixture of Children
What is the probability that a family with 3
children will have at least one child of each
sex?
If X Number of Girls then either family has one
girl and two boys (X 1) or two girls and one
boy (X 2).
P(X 1 or X 2) P(X 1) P(X 2) 3/8
3/8 6/8 3/4
pdf for Number of Girls X
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Expectations for Random Variables
The expected value of a random variable is the
mean value of the variable X in the sample space,
or population, of possible outcomes.
If X is a random variable with possible values
x1, x2, x3, . . . , occurring with probabilities
p1, p2, p3, . . . , then the expected value of X
is calculated as
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Standard Deviation for a Discrete Random Variable
The standard deviation of a random variable is
essentially the average distance the random
variable falls from its mean over the long run.
If X is a random variable with possible values
x1, x2, x3, . . . , occurring with probabilities
p1, p2, p3, . . . , and expected value E(X) m,
then
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Binomial Random Variables
Class of discrete random variables Binomial --
results from a binomial experiment.
Conditions for a binomial experiment
1. There are n trials where n is determined in
advance and is not a random value. 2. Two
possible outcomes on each trial, called success
and failure and denoted S and F. 3. Outcomes
are independent from one trial to the next. 4.
Probability of a success, denoted by p,
remains same from one trial to the next.
Probability of failure is 1 p.
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Examples of Binomial Random Variables
A binomial random variable is defined as Xnumber
of successes in the n trials of a binomial
experiment.
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Finding Binomial Probabilities
for k 0, 1, 2, , n
Example Probability of Two Wins in Three Plays
p probability win 0.2 plays of game are
independent. X number of wins in three plays.
What is P(X 2)?
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Binomial Probability Distribution

• Binomial distribution is based on events in which
  there are only two possible outcomes on each
  occurrence.
• Example Flip a coin 3 times the possible
  outcomes are (heads hits tails misses)
• HHH, HHT, HTT, TTT, TTH, THH, THT, AND HTH

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Binomial Probability Distribution

• Example Flip a coin 3 times the possible
  outcomes are (call heads hits tails misses)

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Binomial Probability Distribution
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Probability Associated with Hits
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Binomial Probability Distribution
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Binomial Probability Distribution
The preceding bar graph is symmetrical this will
always be true for the binomial distribution when
p 0.5.
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Expected Value and Standard Deviation for a
Binomial Random Variable
For a binomial random variable X based on n
trials and success probability p,
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Example Extraterrestrial Life?
50 of large population would say yes if asked,
Do you believe there is extraterrestrial
life? Sample of n 100 is taken. X number
in the sample who say yes is approximately a
binomial random variable.
In repeated samples of n100, on average 50
people would say yes. The amount by which that
number would differ from sample to sample is
about 5.