Probability Distributions Finite Random Variables.
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Transcript of Probability Distributions Finite Random Variables.
Probability Distributions Recall a Random Variable was:
numerical value associated with the outcome of an experiment that was subject to chance
Examples: The number of heads that appear when flipping three coins The sum obtained when two fair dice are rolled Value of an attempted loan workout
Finite Random Variables
Types of random variables: Finite Random Variable:
Can only assume a countable number of values (i.e. they can all be listed)
Examples Flipping a coin three times & recording # of heads Rolling a pair of dice & recording the sum on each roll
Finite Random Variables Types of Random Variables (cont):
Continuous Random Variable: Can assume a whole range of values which cannot be
listed: Examples:
Arrival times of customers using an ATM at the start of the hour
Time between arrivals Length of service at an ATM machine
We’ll first look at finite random variables
Finite Random Variables
Ex: Suppose we flip a coin 3 times and record each flip. Let X be a random variable that records the # of heads we get.
What are the possible values of X?
TTTTTHTHTTHHHTTHTHHHTHHHS ,,,,,,,
Finite Random Variables
X can be 0, 1, 2, or 3
Notice that we can list all possible values of the random variable X
This makes X a finite random variable
Finite Random Variables
Recall Told us the probability the random variable X
(upper case) assumed a certain value x (lower case)
List all possible values of x (lower case) and their probabilities in table
xXP
x P (X=x )
0 0.1251 0.3752 0.3753 0.125
Finite Random Variables
Notice each value of the random variable has one probability associated with it
For a finite random variable we call this the probability mass function
Abbreviated (p.m.f.)x P (X=x )
0 0.1251 0.3752 0.3753 0.125
Finite Random Variables
Why a function? Each value of the random variable has exactly one
probability assigned to it Since this is a function, the following are
equivalent for a finite random variable:
xXPxfX
Finite Random Variables
Like any function, the p.m.f. for a finite random variable has several properties: Domain: discrete numbered values
(e.g. {0, 1, 2,3} )
Range:
Sum: 10 xfX
1 All
x
X xf
Finite Random Variables If we graph a p.m.f., should be a histogram with sum
of all heights equal to 1
Each bar height corresponds to P(X=x)
P(X=x)
0
0.1
0.2
0.3
0.4
0 1 2 3
X=x
P(X=x)
Finite Random Variables
Cumulative Distribution Function Abbreviated c.d.f.
Determines probability of all events occurring up to and including a specific event
xXPxFX
Finite Random Variables
Ex: Find from our coin problem. Do the same for and
Sol:
0XF 7.1XF 2XF
125.000 XPFX
5.0)1(07.17.1 XPXPXPFX
875.021022 XPXPXPXPFX
Finite Random Variables
Notice:
Interval FX(x) = P(X≤x)
(-∞,0) 0
[0,1) 0.125
[1,2) 0.125 + 0.375=0.500
[2,3) 0.125+0.375+0.375=0.875
[3,∞) 0.125+0.375+0.375+0.125=1
Finite Random Variables
The last table is describing a piece-wise function:
3132875.021500.010125.0
00
xifxifxifxif
xif
xFX
Finite Random Variables
The graph of the c.d.f.F(X)
00.1250.25
0.3750.5
0.6250.75
0.8751
1.125
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
X=x
Finite Random Variables Notice for the c.d.f. of a finite random variable
Graph is a step-wise function
Domain is all real #s
Graph never decreases
Approaches 1 as x gets larger
Finite Random Variables
Ex. Use the sample space for rolling two dice to graph . xXP
S
( , ) ( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , ) ( , ) ( , ) ( , ) ( , )
(5, ) (5, ) (5, ) (5, ) (5, ) (5, )
( , ) (
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
1 2 3 4 5 6
6 1 6, ) ( , ) ( , ) ( , ) ( , )2 6 3 6 4 6 5 6 6
Finite Random Variables
Soln.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
1 2 3 4 5 6 7 8 9 10 11 12 13
x P (X =x )
1 0.00002 0.02783 0.05564 0.08335 0.11116 0.13897 0.16678 0.13899 0.1111
10 0.083311 0.055612 0.027813 0.0000
Finite Random Variables
Ex. Find in the previous example. Find in the previous example. Find in the previous example.
Soln.x P (X =x )
1 0.00002 0.02783 0.05564 0.08335 0.11116 0.13897 0.16678 0.13899 0.1111
10 0.083311 0.055612 0.027813 0.0000
3Xf 7Xf 7.2Xf
0556.03 Xf
1667.07 Xf
07.2 Xf
Finite Random Variables
Remember: Cumulative distribution function adds probabilities up to and including a certain value
Ex. Find in the previous example. Find in the previous example. Find in the previous example.
3XF 7XF 7.2XF
Finite Random Variables
Soln.x P (X =x )
1 0.00002 0.02783 0.05564 0.08335 0.11116 0.13897 0.16678 0.13899 0.1111
10 0.083311 0.055612 0.027813 0.0000
0833.00556.00278.00
33
XPFX
5833.01667.0...0278.00
77
XPFX
0278.00278.00
7.27.2
XPFX
Finite Random Variables
• Sample c.d.f.Cummulative Distribution Function
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
0 1 2 3 4 5 6 7 8 9 10
Finite Random Variables Notice the graph starts at height 0
Notice the graph ends at height 1
The graph “steps” to next height
The size of each “step” corresponds to the value of the p.m.f. at that x-value
Finite Random Variables
Binomial Random Variable Special type of finite r.v.
Collection of Bernoulli Trials
Bernoulli Trial is an experiment with only 2 possible outcomes
Each trial is independent
Finite Random Variables Ex: Suppose we flip a biased coin whose probability of
landing heads is 0.7. Let’s say we do this 3 times and record each flip. Let X be a random variable that records the # of heads we get.
This is an example of a binomial random variable
TTTTTHTHTTHHHTTHTHHHTHHHS ,,,,,,,
Finite Random Variables In Excel:
This is the p.m.f
X=x P(X=x)0 0.0271 0.1892 0.4413 0.343
Value of Random Variable
How many times you perform the experiment
Probability of success on each trial
FALSE (p.m.f) ; TRUE (c.d.f)
Finite Random Variables
Ex. Historically, 83% of all students pass a particular class. If there are 34 students in the class, what is the probability that exactly 28 will pass? What is the probability that at least 28 students pass? What is the probability that at most 28 students pass?
Finite Random Variables What is the probability that exactly 28 students pass?
Soln: 0.1760
2828 XPfX
Finite Random Variables What is the probability that at least 28 students pass?
Soln: 1 - 0.3545 0.6455
271
27128128
XFXPXPXP
Finite Random Variables What is the probability that at most 28 students pass?
Soln: 0.5305
xFXP X28
Finite Random Variables
Sample p.m.f.Probability Mass Function
0
0.04
0.08
0.12
0.16
0.2
0 4 8 12 16 20 24 28 32
Finite Random Variables
Mean of a finite random variable same as expected value in project 1 add product of each value and it’s respective
probability
x
XX xfxXE possible all
Finite Random Variables
Ex. Historically, 83% of all students pass a particular class. If there are 34 students in the class, find the mean number of students that will pass.
Soln. Approximately 28.22 students will pass (Two ways this can be found)
Finite Random Variables
Method 1:
Use BINOMDIST to make p.m.f.
Then use:
X=x P(X=x) x*P(X=x)0 6.84326E-27 01 1.13598E-24 1.14E-242 9.15134E-23 1.83E-223 4.76587E-21 1.43E-204 1.80332E-19 7.21E-195 5.28267E-18 2.64E-176 1.24661E-16 7.48E-167 2.43455E-15 1.7E-148 4.01164E-14 3.21E-139 5.65825E-13 5.09E-12
10 6.90639E-12 6.91E-1111 7.35696E-11 8.09E-1012 6.88453E-10 8.26E-0913 5.68831E-09 7.39E-0814 4.16585E-08 5.83E-0715 2.71188E-07 4.07E-0616 1.5723E-06 2.52E-0517 8.12806E-06 0.00013818 3.74794E-05 0.00067519 0.000154095 0.00292820 0.000564259 0.01128521 0.001836607 0.03856922 0.005298661 0.11657123 0.013497356 0.31043924 0.030203643 0.72488725 0.058985939 1.47464826 0.099688905 2.59191227 0.144212272 3.89373128 0.176023803 4.92866629 0.177809034 5.15646230 0.144687744 4.34063231 0.091150533 2.82566732 0.041721476 1.33508733 0.012345392 0.40739834 0.001772781 0.060275
E(X) 28.22
x
XX xfxXE possible all