hudm4122 probability and statistical inference...probability and statistical inference february 23,...
TRANSCRIPT
-
HUDM4122Probability and Statistical Inference
February 23, 2015
-
In the last class
• We studied Bayes’ Theorem and the Law ofTotal Probability
-
Any questions or comments?
-
Today
• Chapter 4.8 in Mendenhall, Beaver, & Beaver
• Probability Distributions• Random Variables
-
Random Variable
• “A variable x is a random variable if the valuethat it assumes, corresponding to theoutcome of an experiment, is a chance orrandom event.” – MBB, p.158
• “A random variable is a variable whose valueis subject to variations due to chance.” --Wikipedia
-
Random Variable
• It is not that the variable can have any value atrandom
• But that the variable’s value comes fromrandom sampling
-
These are random variables
• A coin flip’s result• Number of times a randomly selected student
is sent to the principal’s office• NY Regents Exam score for a randomly
selected student in NY State• Number of people on the subway at a
randomly selected time
-
These are not random variables
• I flip a biased coin that gives 100% heads• The temperature setting on your oven – you
set it yourself
• These values are not subject to chance
-
A random variable’s value
• You can never say for sure what it’s going tobe
• There is a certainly probability that it will havecertain values
-
Questions? Comments?
-
Reminder
• Discrete variable– Can have limited number of values
• Continuous/numerical variable– Can have infinite number of values
-
Which of these are Discrete Variables?
• Number of heads in 3 coin flips• Sum of rolling two 6-sided dice• Temperature outside• How late is 2 train• Height of a person, rounded to closest inch• Number of times a randomly selected student
is sent to the principal’s office
-
Probability Distribution
• A probability distribution for random variableX– gives the possible values of X, x1…xn– And the probability p(xi) associated with each
value of X
-
Probability Distribution
• A probability distribution for random variableX– gives the possible values of X, x1…xn– And the probability p(xi) associated with each
value of X
– Each value of X is mutually exclusive– The sum of p(xi) adds to 1
-
Example
• I flip a coin twice• The number of heads can be 0, 1, or 2
• TT: 0• TH:1• HT:1• HH:2
-
Example
• I flip a coin twice• The number of heads can be 0, 1, or 2
• TT: 0• TH:1• HT:1• HH:2
x P(x)012
-
Example
• I flip a coin twice• The number of heads can be 0, 1, or 2
• TT: 0• TH:1• HT:1• HH:2
x P(x)0 1/41 2/42 1/4
-
Example
• I flip a coin twice• The number of heads can be 0, 1, or 2
• TT: 0• TH:1• HT:1• HH:2
x P(x)0 1/41 1/22 1/4
-
Example
• I flip a coin twice• The number of heads can be 0, 1, or 2
• TT: 0• TH:1• HT:1• HH:2
x P(x)0 0.251 0.52 0.25
-
Probability Histogram
x P(x)0 0.251 0.52 0.25
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2
p(x)
x
-
You try it
• I flip a coin three times• What is the probability distribution on the
number of heads?
-
You try it:What is the Probability Distribution?
• I collected the following data on thetemperature in NYCDay Temp Day Temp
1 Cold 9 Cold2 Cold 10 Cold3 Freezing 11 Cold4 Cold 12 Not That Bad5 Cold 13 Cold6 Freezing 14 F’ing Freezing7 F’ing Freezing 15 F’ing Freezing8 Freezing 16 Cold
-
Note that probability distributions canhave many values
00.10.20.30.40.50.60.70.80.9
1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
p(x)
x
-
Later in the semester
• We’ll talk about probability distributions forcontinuous variables
-
Questions? Comments?
-
Expected Value ofProbability Distribution
• The value you can expect to get on average ifyou re-run your experiment many times
• Take each value, multiply it by its probability• Add those together
• E(x) = ∑ ∗ ( )
-
Expected Value ofProbability Distribution
• The value you can expect to get on average if youre-run your experiment many times
• Take each value, multiply it by its probability• Add those together
• E(x) = ∑ ∗ ( )• This is also called the mean of the random
variable, µ
-
Example
• 0(0.25) + 1(0.5) + 2(0.25)
x P(x)0 0.251 0.52 0.25
-
Example
• 0 + 0.5 + 0.5 = 1
x P(x)0 0.251 0.52 0.25
-
You can expect 1 head on average ifyou flip a coin twice, infinite times
• 0 + 0.5 + 0.5 = 1
x P(x)0 0.251 0.52 0.25
-
You Try Itx P(x)0 0.11 0.52 0.23 0.2
-
You Try It
• 0(0.1)+1(0.5)+2(0.2)+3(0.2)
x P(x)0 0.11 0.52 0.23 0.2
-
You Try It
• 0+0.5+0.4+0.6= 1.5
x P(x)0 0.11 0.52 0.23 0.2
-
You Try It
• My dad’s barber has played the lottery everyday for 40 years (a.k.a. 14,600 times), at $1 aticket
• One time he won $1000• What is the expected value for playing the
lottery?
-
You Try It
• My dad’s barber has played the lottery everyday for 40 years (a.k.a. 14,600 times), at $1 aticket
• One time he won $1000• What is the expected value for playing the
lottery?
• (1/14600)(1000) + (14599/14600)(-1)
-
You Try It
• My dad’s barber has played the lottery everyday for 40 years (a.k.a. 14,600 times), at $1 aticket
• One time he won $1000• What is the expected value for playing the
lottery?
• (1/14600)(1000) + (14599/14600)(-1)=0.068 – 0. 999 = $-0.931
-
Questions? Comments?
-
Variance of Discrete Random Variable
•∑ − ( )
-
Standard Deviation ofDiscrete Random Variable
• ∑ − ( )
-
Example: SD of Random Variable
x P(x)0 0.251 0.52 0.25
-
Example: SD of Random Variable
x P(x)0 0.251 0.52 0.25
• Recall: µ = 1
-
Example: SD of Random Variablex P(x)0 0.251 0.52 0.25
• Recall: µ = 1 − ( )
-
Example: SD of Random Variablex P(x)0 0.251 0.52 0.25
• Recall: µ = 1
• 0 − 1 (0.25) + 1 − 1 .5+ 2 − 1 .25
-
Example: SD of Random Variablex P(x)0 0.251 0.52 0.25
• Recall: µ = 1
• 1(0.25) + 0 .5+1 .25
-
Example: SD of Random Variablex P(x)0 0.251 0.52 0.25
• Recall: µ = 1• 0.25 + 0.25
-
Example: SD of Random Variablex P(x)0 0.251 0.52 0.25
• Recall: µ = 1• 0.25 + 0.25• 0.707
-
You Try It: SD of random variable
• µ = 1.5
x P(x)0 0.11 0.52 0.23 0.2
-
Any last comments or questions forthe day?
-
If there’s time:Do this in solver-explainer pairs
• Practice with extended version of Bayes Rule
• ( | ) = )∑ ( | )
-
Example
• There are three professors who teachHUDM4122. Let’s call them A, B, and C
• P(student is in prof A’s class) = P(A) = 0.4• P(student is in prof B’s class) = P(B) = 0.3• P(student is in prof C’s class) = P(C) = 0.1
-
Example
• P(student is in prof A’s class) = P(A) = 0.4• P(student is in prof B’s class) = P(B) = 0.3• P(student is in prof C’s class) = P(C) = 0.1
• P(learned stats|A)=0.8• P(learned stats|B)=0.6• P(learned stats|C)=0.4
-
What is P(A | learned stats)?
• P(student is in prof A’s class) = P(A) = 0.4• P(student is in prof B’s class) = P(B) = 0.3• P(student is in prof C’s class) = P(C) = 0.1
• P(learned stats|A)=0.8• P(learned stats|B)=0.6• P(learned stats|C)=0.4
-
Upcoming Classes
• 2/25 Binomial Probability Distribution– Ch. 5-2– HW 4 due
• 2/28 Normal Probability Distribution
-
Homework 4
• Due in 2 days• In the ASSISTments system