1. HUDM4122 Probability and Statistical Inference
February 23, 2015

2. In the last class
We studied Bayes’ Theorem and the Law of Total Probability

3. Any questions or comments?

4. Today Chapter 4.8 in Mendenhall, Beaver, & Beaver
Probability Distributions
Random Variables

5. Random Variable
“A variable x is a random variable if the value that it assumes, corresponding to the outcome of an experiment, is a chance or random event.” – MBB, p.158
“A random variable is a variable whose value is subject to variations due to chance.” -- Wikipedia

6. Random Variable
It is not that the variable can have any value at random
But that the variable’s value comes from random sampling

7. 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

8. 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

9. A random variable’s value
You can never say for sure what it’s going to be
There is a certainly probability that it will have certain values

10. Questions? Comments?

11. Reminder Discrete variable Continuous/numerical variable
Can have limited number of values
Continuous/numerical variable
Can have infinite number of values

12. 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

13. Probability Distribution
A probability distribution for random variable X
gives the possible values of X, x1…xn
And the probability p(xi) associated with each value of X

14. Probability Distribution
A probability distribution for random variable X
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

15. Example I flip a coin twice The number of heads can be 0, 1, or 2
TT: 0
TH:1
HT:1
HH:2

16. Example x P(x) 1 2 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) 1 2

17. Example x P(x) 1/4 1 2/4 2 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)
1/4 1
2/4 2

18. Example x P(x) 1/4 1 1/2 2 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)
1/4 1
1/2 2

19. Example x P(x) 0.25 1 0.5 2 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)
0.25 1
0.5 2

20. Probability Histogramx
P(x)
0.25 1
0.5 2

21. You try it I flip a coin three times
What is the probability distribution on the number of heads?

22. You try it: What is the Probability Distribution?
I collected the following data on the temperature in NYC
Day
Temp 1
Cold 9 2 10 3
Freezing 11 4 12
Not That Bad 5 13 6 14
F’ing Freezing 7 15 8 16

23. Note that probability distributions can have many values

24. Later in the semester
We’ll talk about probability distributions for continuous variables

25. Questions? Comments?

26. Expected Value of Probability Distribution
The value you can expect to get on average if you re-run your experiment many times
Take each value, multiply it by its probability
Add those together
E(x) = 𝑥∗𝑝(𝑥)

27. Expected Value of Probability Distribution
The value you can expect to get on average if you re-run your experiment many times
Take each value, multiply it by its probability
Add those together
E(x) = 𝑥∗𝑝(𝑥)
This is also called the mean of the random variable, m

28. Example
0(0.25) + 1(0.5) + 2(0.25) x
P(x)
0.25 1
0.5 2

29. Example
= 1 x
P(x)
0.25 1
0.5 2

30. You can expect 1 head on average if you flip a coin twice, infinite times
= 1 x
P(x)
0.25 1
0.5 2

31. You Try It x
P(x)
0.1 1
0.5 2
0.2 3

32. You Try It x
P(x)
0.1 1
0.5 2
0.2 3
0(0.1)+1(0.5)+2(0.2)+3(0.2)

33. You Try It x
P(x)
0.1 1
0.5 2
0.2 3
= 1.5

34. You Try It
My dad’s barber has played the lottery every day for 40 years (a.k.a. 14,600 times), at $1 a ticket
One time he won $1000
What is the expected value for playing the lottery?

35. You Try It
My dad’s barber has played the lottery every day for 40 years (a.k.a. 14,600 times), at $1 a ticket
One time he won $1000
What is the expected value for playing the lottery?
(1/14600)(1000) + (14599/14600)(-1)

36. You Try It
My dad’s barber has played the lottery every day for 40 years (a.k.a. 14,600 times), at $1 a ticket
One time he won $1000
What is the expected value for playing the lottery?
(1/14600)(1000) + (14599/14600)(-1)= – = $-0.931

37. Questions? Comments?

38. Variance of Discrete Random Variable
𝑥−𝜇 2 𝑝(𝑥)

39. Standard Deviation of Discrete Random Variable
𝑥−𝜇 2 𝑝(𝑥)

40. Example: SD of Random Variable
P(x)
0.25 1
0.5 2

41. Example: SD of Random Variable
P(x)
0.25 1
0.5 2
Recall: m = 1

42. Example: SD of Random Variable
P(x)
0.25 1
0.5 2
Recall: m = 1
𝑥−𝜇 2 𝑝(𝑥)

43. Example: SD of Random Variable
P(x)
0.25 1
0.5 2
Recall: m = 1
0−1 2 (0.25)+ 1− −

44. Example: SD of Random Variable
P(x)
0.25 1
0.5 2
Recall: m = 1
1(0.25)

45. Example: SD of Random Variable
P(x)
0.25 1
0.5 2
Recall: m = 1

46. Example: SD of Random Variable
P(x)
0.25 1
0.5 2
Recall: m = 1
0.707

47. You Try It: SD of random variablex
P(x)
0.1 1
0.5 2
0.2 3
m = 1.5

48. Any last comments or questions for the day?

49. If there’s time: Do this in solver-explainer pairs
Practice with extended version of Bayes Rule
𝑃( 𝑆 𝑖 |𝐴) = 𝑃 𝑆 𝑖 𝑃 𝐴 𝑆 𝑖 ) 𝑗=1 𝑘 𝑃 𝑆 𝑗 𝑃(𝐴| 𝑆 𝑗 )

50. Example
There are three professors who teach HUDM4122. 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

51. 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

52. 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

53. Upcoming Classes 2/25 Binomial Probability Distribution
Ch. 5-2
HW 4 due
2/28 Normal Probability Distribution

54. Homework 4
Due in 2 days
In the ASSISTments system