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Lecture 6

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Statistics

01:960:401

Micheal Miniere

Summer

Description

Chapter 5 (Probability Distributions)
Sections 1 and 2:
Definition: (Random Variables)
t, th
denoted by X,can assume any one of these numbers
In mathematical language we say that a random variable X is a real-valued function defined
on a sample space (see page 173 in your text). Note, little
usually denotes a particular
value that the random variable X can assume, and the probability that a particular value x
occurs will be denoted by P(x
Note: The values of a random variable represent every simple event in the sample space of an
experiment. Moreover, the word random is used because we don't know whatvalue the
variable takes on until after the experiment is conducted.
We will do some examples in class
Definition: A Discrete Random Variable (Chapter Five) assumes a finite number of values or
infinitely many values that can be arranged in a sequence, i.e., those values usually associated
with a counting process
Definition: A Continuous Random Variable (Chapter Six) can assume all the values in an interval
I number li
Definition: A probability distribution of a discrete random variable is a list of all the possible
values that a random variable assumes along with their corresponding probabilities.
There are two properties that a probability distribution of a discrete random variable must
1. Y POx)-1, for all values of x
2. 0s P
for each value of
We will do examples in class.
ec hurt lo Chapter 5
Random ar
Ilip a coin onll
et x be the g heads possible 0,l
er roll a die anu
et Y b Hue g possible outlohnus
discrete variable: list every H the problrm can take on
Rumbability tribution at a
scrtk Random Variable
s a list of au possibl Valul S that lan assume alo
orresponding probability
with the
ex flip g coin and let X be the of hads possible
0 i2 b probability distribution
l obabity histogram
ex roll a die and ltt X be Hu H that llmlS up
I anta has to
3 5 b
har Lunidorm, probability histogram
Chapter 5 (Probability Distributions) Sections 1 and 2: Definition: (Random Variables) t, th denoted by X,can assume any one of these numbers In mathematical language we say that a random variable X is a real-valued function defined on a sample space (see page 173 in your text). Note, little usually denotes a particular value that the random variable X can assume, and the probability that a particular value x occurs will be denoted by P(x Note: The values of a random variable represent every simple event in the sample space of an experiment. Moreover, the word random is used because we don't know whatvalue the variable takes on until after the experiment is conducted. We will do some examples in class Definition: A Discrete Random Variable (Chapter Five) assumes a finite number of values or infinitely many values that can be arranged in a sequence, i.e., those values usually associated with a counting process Definition: A Continuous Random Variable (Chapter Six) can assume all the values in an interval I number li Definition: A probability distribution of a discrete random variable is a list of all the possible values that a random variable assumes along with their corresponding probabilities. There are two properties that a probability distribution of a discrete random variable must 1. Y POx)-1, for all values of x 2. 0s P for each value of We will do examples in class. ec hurt lo Chapter 5 Random ar Ilip a coin onll et x be the g heads possible 0,l er roll a die anu et Y b Hue g possible outlohnus discrete variable: list every H the problrm can take on Rumbability tribution at a scrtk Random Variable s a list of au possibl Valul S that lan assume alo orresponding probability with the ex flip g coin and let X be the of hads possible 0 i2 b probability distribution l obabity histogram ex roll a die and ltt X be Hu H that llmlS up I anta has to 3 5 b har Lunidorm, probability histogramChapter 5 Section 3:
Definition: The mean of a discrete random variable
X s given by:
Definition: The standard deviation of a discrete random variable is given by
o -H P
o sigma blic entire population (disurtle)
Now the short-cut version i
by
Note, gives a measure of how much the probability distribution is spread about the mean of
the discrete random variable
X. Also the Empirical Rule and Chebyshev's Rule also apply
probability distributions.
We will find out in class that the mean for the random variable
X
nts the
y d
3.5 Wh
that the average of all the observed values of the random variable will be approximately equal
to 3.5. That is why the mean of a random variable is often called the expected value or
mathematical expectation of a random variable; and
Definition: The expected value of a discrete random variable is
Consider the numbers game (started many years ago by organized crime and now run legally by
many organized governments). You bet $1 that a 3-digit number of your choice will win out of
get 500
bet plus $499
1000
000
999
499
find
his g
0.50
$0.50. T
bet, y
pect
on this game is 50%
Defi
A g
g eq
the case, then there is no advantage to you or the house. If E(x) is positive, the game i
d if E
negative, the game is unfavorable to the
Example: In a game of Roulette, you bet $1. If you win you get $36, i.e., your bet plus $35
ble X
assume
amount that you win. Then 35
or
l We will find the expected value in class
Lecturt b l Chaper 5 on
2,2 2,3 2,4 2,5 2,b
possibl
2 33 34 3,5 3
rolling 2 diu
L 4,2 43 4,4 45 4,
5.1 5,2 5,3 5,4 55 5,
probability distributian probability histogram
a thu b
53
36
H 2
shilt vallus over to lit undar
he bar
calculator: put x's in Li put correspond px)'s in Li
go to L3 highlight L3 Li XL23 enttr
1- var stats L37 enttr sample in a
-var stats List La Preg L2 calculath z info
in given lov disutte applies to a papulation b/c
all possible outlomos art accounta or
pg 23
Chapter 5 Section 3: Definition: The mean of a discrete random variable X s given by: Definition: The standard deviation of a discrete random variable is given by o -H P o sigma blic entire population (disurtle) Now the short-cut version i by Note, gives a measure of how much the probability distribution is spread about the mean of the discrete random variable X. Also the Empirical Rule and Chebyshev's Rule also apply probability distributions. We will find out in class that the mean for the random variable X nts the y d 3.5 Wh that the average of all the observed values of the random variable will be approximately equal to 3.5. That is why the mean of a random variable is often called the expected value or mathematical expectation of a random variable; and Definition: The expected value of a discrete random variable is Consider the numbers game (started many years ago by organized crime and now run legally by many organized governments). You bet $1 that a 3-digit number of your choice will win out of get 500 bet plus $499 1000 000 999 499 find his g 0.50 $0.50. T bet, y pect on this game is 50% Defi A g g eq the case, then there is no advantage to you or the house. If E(x) is positive, the game i d if E negative, the game is unfavorable to the Example: In a game of Roulette, you bet $1. If you win you get $36, i.e., your bet plus $35 ble X assume amount that you win. Then 35 or l We will find the expected value in class Lecturt b l Chaper 5 on 2,2 2,3 2,4 2,5 2,b possibl 2 33 34 3,5 3 rolling 2 diu L 4,2 43 4,4 45 4, 5.1 5,2 5,3 5,4 55 5, probability distributian probability histogram a thu b 53 36 H 2 shilt vallus over to lit undar he bar calculator: put x's in Li put correspond px)'s in Li go to L3 highlight L3 Li XL23 enttr 1- var stats L37 enttr sample in a -var stats List La Preg L2 calculath z info in given lov disutte applies to a papulation b/c all possible outlomos art accounta or pg 23Chapter 5 Section 5:
Now we consider a common type of
experiment consisting of only two outcomes
A Binomial Experiment is an experiment that meets four conditions:
1. There must be a fixed number of trials
2. There are only two possible outcomes for each trial
3. Each trial must be dependent.
A. The probabilities must remain constant for each trial.
Notation:
and F (success and failure) denote the two outcomes
n denotes the fixed number of trials
p denotes the probability of a success in one of the n trials
denotes the probability of a failure in one of the n trials. Note q -1-p
q Hence POST
POF
P 9
The random variable X denotes the number of successes in n trials of a binomial experiment
and is called the binomial random variable. PLX
or just POx) denotes the probability of
getting exactly x successes among the n tria
Binomial Probability Formula
The probability of getting exactly x successes among n trials of a binomial experiment, where
p is the probability of a success, is
P q
We will do the following examples in
class
Example: What is the probability of getting
ctly
seven heads in 15 flips of a coin?
Example: What is the probability of rolling exactly six one's among 17 rolls of a die?
Example: What is the probability of getting at least six heads in ten flips of a coin?
Example: What is
the probability of getting at most eight heads in 15 flips of a coin?
We now consider the Binomial Distribution.
The Binomial Distribution is a complete list of all the probabilities of getting x successes in n
trials of a binomial experiment. We will find the probability distribution, along with the
probability histogram, for the random variable X, where X is the number of heads possible in
ten flips of a coin. This will be done in class
Note: As the number of trials in a binomial experiment increases, the distribution of the
Binomial Distribution begins to assume a bell shape. As a rule of thumb, if np and nq are
greater than or equal to ten, the binomial probability distribution will be approximately bell
shaped. See page 242 in your text
Finally recall that the mean and standard deviation for any probability distribution is
POx) or short-cut version or
but for the special case of the B
Distribution, we have a simpler version of these:
Au-np (also expected value)
Example: consider the pervious example pertaining to the probability distribution for the
random variable ere
X is the number of heads possible in ten flips of a coin
np 10x0.5
10x0.5x0.5 V2.5 1.58
Chapter 5 Section 5: Now we consider a common type of experiment consisting of only two outcomes A Binomial Experiment is an experiment that meets four conditions: 1. There must be a

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