1. CS/EE 144 The ideas behind our networked world
A probability refresher

2. What kind of problems should you be able to solve?
1. A total of 𝑟 balls are placed, one at a time, into 𝑘 boxes. Each ball is placed independently into box 𝑖 with probability 𝑝 𝑖 (with 𝑝 1 + 𝑝 2 +…+ 𝑝 𝑘 =1). What is the expected number of empty boxes? What is the expected number of boxes containing at least 2 balls? 2. There are 𝑟 red balls and 𝑏 black balls in a box. We take out the balls one at a time randomly. What is the expected number of red balls that precede all black balls? 3. A biased coin has a probability 𝑝 of showing heads when tossed. The coin is tossed repeatedly until a head occurs. What is the expected number of tosses required to obtain a head? Solve this without calculating an infinite sum.

3. Outline Probability space Random variables Expectation
Indicator random variable
Conditional probability and conditional expectation
Important discrete random variables
Important continuous random variables

4. Experiment: toss a fair coin three times
What are the outcomes and probabilities?
Probability that there are two heads?
Expected number of heads?
Is number of heads independent of number of tails? H T
HHH
HHT
HTH
HTT
TTH
THH
THT
TTT

5. Probability space A probability space consists of (Ω,𝑃):
Ω is set of all possible outcomes.
𝑃:Ω→ 0,1 is an assignment of probabilities to outcomes. Note that 𝜔∈Ω 𝑃 𝜔 =1
e.g. three coin tosses: Ω= 𝐻𝐻𝐻,𝐻𝐻𝑇,𝐻𝑇𝐻,𝐻𝑇𝑇,𝑇𝐻𝐻,𝑇𝐻𝑇,𝑇𝑇𝐻,𝑇𝑇𝑇 𝑃 𝜔 = 1 8 , 𝜔∈Ω
Typically interested in collections of outcomes, e.g. outcomes with at least two heads. We refer to such collections as events.
e.g. 𝐸= 𝐻𝐻𝐻,𝐻𝐻𝑇,𝐻𝑇𝐻,𝑇𝐻𝐻 is the event that there are at least two heads, 𝑃 𝐸 = 1 2

6. Technically, more machinery is needed to define a probability space, you can learn this in a Math course on probability theory

7. Useful properties
Partition property: given any partition of events 𝐸 1 , 𝐸 2 , …, 𝐸 𝑛 of Ω 𝑖=1 𝑛 𝑃( 𝐸 𝑖 ) =1
Union bound: given any events 𝐸 1 , 𝐸 2 , …, 𝐸 𝑛 𝑃 𝑖=1 𝑛 𝐸 𝑖 ≤ 𝑖=1 𝑛 𝑃( 𝐸 𝑖 )

8. Independence
Definition: Two events 𝐸 1 and 𝐸 2 are independent ⟺𝑃 𝐸 1 ∩ 𝐸 2 =𝑃 𝐸 1 𝑃( 𝐸 2 )
e.g. coin toss experiment: 𝐴 denotes the event that all the tosses have the same faces 𝐵 denotes the event that there is at most one head 𝐶 denotes the event that there is one head and one tail 𝑃 𝐴 = 1 4 𝑃 𝐵 = 1 2 𝑃 𝐶 = 3 4 Are 𝐴 and 𝐵 independent? Yes. 𝑃 𝐴∩𝐵 = 1 8 Are 𝐵 and 𝐶 independent? Yes. 𝑃 𝐵∩𝐶 = 3 8 Are 𝐴 and 𝐶 independent? No. 𝑃 𝐴∩𝐶 =0
HHH
HHT
HTH
HTT
TTH
THH
THT
TTT

9. Set of possible outcomes Ω
Random variables
A random variable 𝑋 is a function mapping Ω to ℝ, i.e. 𝑋:Ω→ℝ e.g. 𝑋= number of heads in three coin tosses 𝑌= number of tails in three coin tosses 𝑍= 1 if all three tosses show the same face and 0 otherwise 𝑃 𝑋=𝑘 ≔𝑃 {𝜔∈Ω: 𝑋 𝜔 =𝑘} e.g. coin toss experiment: 𝐴 denotes the event that all the tosses have the same faces, 𝑋=3 ∪{𝑋=0} 𝐵 denotes the event that there is at most one head, 𝑋≤1 𝐶 denotes the event that there is one head and one tail, 1≤𝑋≤2
Set of possible outcomes Ω ℝ 𝑋

10. Independence of random variables
How might we extend the notion of independence of events to random variables? Might desire this: if random variables 𝑋 and 𝑌 are independent, then for any 𝐴⊂ℝ and 𝐵⊂ℝ, the events {𝑋∈𝐴} and {𝑌∈𝐵} are independent. It turns out that we do not have to check all possible subsets 𝐴 and 𝐵!
Definition: Two random variables 𝑋 and 𝑌 are independent ⟺ for every 𝑥 and 𝑦, the events 𝑋≤𝑥 and {𝑌≤𝑦} are independent.

11. Quiz
Let 𝑋 and 𝑌 be the number of heads and tails respectively in the coin toss experiment.
Are 𝑋 and 𝑌 identical?
No. They are different functions!
Are the probability distributions of 𝑋 and 𝑌 identical?
Yes. In other words, “𝑋 and 𝑌 are identically distributed”.
Are 𝑋 and 𝑌 independent?
No. For instance, the events 𝑋=2 and {𝑌=2} are not independent since we cannot simultaneously have 2 heads and 2 tails.

12. Discrete vs. continuous random variables
A discrete random variable takes values in ℤ. It is described by a probability mass function (pmf). 𝑃 𝑋=𝑘 = 𝑝 𝑋 𝑘
Properties: 𝑝 𝑋 𝑘 ∈ 0,1 , 𝑘∈ℤ 𝑝 𝑋 (𝑘) =1
A continuous random variable takes values in an interval (or a collection of intervals) in ℝ. It is described by a probability density function (pdf). 𝑃 𝑎≤𝑋≤𝑏 = 𝑎 𝑏 𝑓 𝑋 𝑠 𝑑𝑠 Often, the cumulative distribution function (cdf) is also used 𝐹 𝑋 𝑥 ≔𝑃 𝑋≤𝑥 = −∞ 𝑥 𝑓 𝑋 𝑠 𝑑𝑠 Properties: 𝑓 𝑋 (𝑥)≥0, 𝐹 𝑋 −∞ =0, 𝐹 𝑋 +∞ =1, 𝑓 𝑋 𝑥 = 𝑑 𝑑𝑥 𝐹 𝑋 𝑥

13. Expectation
Long-run average of the values taken by the random variable when the same experiment is performed repeatedly. Also called the “mean” or the “first moment”. 𝐸 𝑋 = 𝑘∈ℤ 𝑘 𝑝 𝑋 (𝑘) 𝐸 𝑋 = −∞ +∞ 𝑠 𝑓 𝑋 𝑠 𝑑𝑠
If 𝑐 is a constant, then 𝐸 𝑐 =𝑐.
Linearity of expectation 𝐸 𝑎𝑋+𝑏𝑌 =𝑎𝐸 𝑋 +𝑏𝐸[𝑌]
𝑋 and 𝑌 independent ⟹ 𝐸 𝑋𝑌 =𝐸 𝑋 𝐸[𝑌]

14. Indicator random variable
The indicator random variable for the event 𝐸 is given by 𝟏 𝐸 𝜔 = 1, 𝑖𝑓 𝜔∈𝐸 0, 𝑖𝑓 𝜔∉𝐸
Important fact: 𝐸 𝟏 𝐸 =1⋅𝑃 𝐸 +0⋅𝑃 𝐸 𝑐 =𝑃(𝐸)
Indicators are tremendously useful for counting expected number of occurrences! Consider coin toss experiment. Let 𝐸 𝑖 be the event that the 𝑖 𝑡ℎ toss results in a head. Then the total number of heads is given by 𝟏 𝐸 1 + 𝟏 𝐸 2 + 𝟏 𝐸 3 . The expected number of heads is 𝐸 𝟏 𝐸 1 + 𝟏 𝐸 2 + 𝟏 𝐸 3 =𝐸[ 𝟏 𝐸 1 ]+𝐸[ 𝟏 𝐸 2 ]+ 𝐸[𝟏 𝐸 3 ] =𝑃 𝐸 1 +𝑃 𝐸 2 +𝑃 𝐸 3 = 3 2
Note: We did not require 𝐸 𝑖 to be independent or identically distributed. An example will be given in the next two slides.
Linearity

15. A total of 𝑟 balls are placed, one at a time, into 𝑘 boxes
A total of 𝑟 balls are placed, one at a time, into 𝑘 boxes. Each ball is placed independently into box 𝑖 with probability 𝑝 𝑖 (with 𝑝 1 + 𝑝 2 +…+ 𝑝 𝑘 =1).
What is the expected number of empty boxes?
What is the expected number of boxes containing at least 2 balls?

16. 2. There are 𝑟 red balls and 𝑏 black balls in a box
2. There are 𝑟 red balls and 𝑏 black balls in a box. We take out the balls one at a time randomly. What is the expected number of red balls that precede all black balls?

17. Conditional probability
Coin toss experiment: suppose your friend tells you that there is at most one head, what is the probability that all three tosses have the same faces?
Definition: Given two events 𝐴 and 𝐵 such that 𝑃 𝐵 >0, the conditional probability of 𝐴 given 𝐵 is 𝑃 𝐴 𝐵 = 𝑃(𝐴∩𝐵) 𝑃(𝐵)
Events 𝐴 and 𝐵 are independent ⟺ 𝑃 𝐴 𝐵 =𝑃(𝐴).
Law of total probability: given any partition of events 𝐸 1 , 𝐸 2 , …, 𝐸 𝑛 of Ω where 𝑃 𝐸 𝑖 >0 𝑃 𝐴 = 𝑖=1 𝑛 𝑃 𝐸 𝑖 𝑃(𝐴| 𝐸 𝑖 )

18. Conditional probability
e.g. coin toss experiment 𝐴 denotes the event that all the tosses have the same faces 𝐵 denotes the event that there is at most one head 𝐶 denotes the event that there is one head and one tail 𝑃 𝐴 = 1 4 𝑃 𝐵 = 1 2 𝑃 𝐶 = 3 4 𝑃 𝐴|𝐵 = 𝑃 𝐴∩𝐵 𝑃 𝐵 = 1/8 1/2 = 1 4 =𝑃 𝐴 𝑃 𝐴|𝐶 = 𝑃 𝐴∩𝐶 𝑃 𝐶 = 0 3/4 =0≠𝑃 𝐴
HHH
HHT
TTH
HTT
HTH
THH
THT
TTT

19. Conditional expectation
Long-run average of the values taken by a random variable conditioned on the occurrence of a particular event. 𝐸 𝑋|𝐴 = 𝑘∈ℤ 𝑘 𝑝 𝑋 (𝑘|𝐴) 𝐸 𝑋|𝐴 = −∞ +∞ 𝑠 𝑓 𝑋 𝑠|𝐴 𝑑𝑠
𝑋 and 𝑌 are independent ⟹ 𝐸 𝑋|𝑌=𝑦 =𝐸 𝑋
Law of total expectation: given any partition of events 𝐸 1 , 𝐸 2 , …, 𝐸 𝑛 of Ω where 𝑃 𝐸 𝑖 >0 𝐸 𝑋 = 𝑖=1 𝑛 𝑃 𝐸 𝑖 𝐸 𝑋 𝐸 𝑖

20. Conditional expectation
Law of total expectation allows us to break a problem into pieces! A coin has a probability 𝑝 of showing heads. Let 𝑁 be a discrete random variable taking non-negative values. Assume that 𝑁 is independent of our coin tosses. What is the expected number of heads that show up if we toss this coin 𝑁 times? 𝑋 𝑖 = 1, 𝑖 𝑡ℎ 𝑡𝑜𝑠𝑠 𝑖𝑠 𝑎 ℎ𝑒𝑎𝑑 0, 𝑖 𝑡ℎ 𝑡𝑜𝑠𝑠 𝑖𝑠 𝑎 𝑡𝑎𝑖𝑙 𝑆= 𝑋 1 + 𝑋 2 +…+ 𝑋 𝑁 𝐸 𝑆 =𝐸 𝐸 𝑆|𝑁 =𝐸 𝐸 𝑋 1 + 𝑋 2 +…+ 𝑋 𝑁 |𝑁 =𝐸 𝐸 𝑋 1 𝑁 +𝐸 𝑋 2 𝑁 +…+𝐸 𝑋 𝑁 𝑁 =𝐸 𝑝+𝑝+…+𝑝 =𝐸 𝑁𝑝 =𝐸 𝑁 𝑝
Indicator random variable
Law of total expectation
Linearity
Independence

21. 3. A biased coin has a probability 𝑝 of showing heads when tossed
3. A biased coin has a probability 𝑝 of showing heads when tossed. The coin is tossed repeatedly until a head occurs. What is the expected number of tosses required to obtain a head? Solve this without calculating an infinite sum.

22. Know the following random variables well
Bernoulli
Binomial
Geometric
Poisson
Uniform
Exponential
Normal/Gaussian
When they arise
Distribution functions
Moments
How they relate to each other

23. Bernoulli random variable
A biased coin has a probability 𝑝 of showing heads when tossed. Let 𝑋 take the value 1 if the outcome is heads and 0 if the outcome is tails.
Often, the events corresponding to 𝑋=1 and 𝑋=0 are referred to as “success” and “failure”.
𝑝 𝑋 𝑘 = 1−𝑝, 𝑖𝑓 𝑘=0 𝑝, 𝑖𝑓 𝑘= 𝐸 𝑋 =𝑝 𝑉𝑎𝑟 𝑋 =𝑝(1−𝑝)
Notation: 𝑋∼𝐵𝑒𝑟𝑛𝑜𝑢𝑙𝑙𝑖(𝑝)

24. Binomial random variable
A biased coin has a probability 𝑝 of showing heads when tossed. Let 𝑋 denote the total number of heads obtained over 𝑛 tosses.
Perform 𝑛 independent Bernoulli trials 𝑋 1 , 𝑋 2 , …, 𝑋 𝑛 and let 𝑋= 𝑋 1 + 𝑋 2 +…+ 𝑋 𝑛 denote the total number of successes.
𝑝 𝑋 𝑘 = 𝑛 𝑘 𝑝 𝑘 1−𝑝 𝑛−𝑘 , 𝑘=0, 1, …, 𝑛 𝐸 𝑋 =𝑛𝑝 𝑉𝑎𝑟 𝑋 =𝑛𝑝(1−𝑝)
Notation: 𝑋∼𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙(𝑛,𝑝)
Exercise: use indicator variables and linearity of expectation to derive the expectation of the binomial random variable

25. Geometric random variable
A biased coin has a probability 𝑝 of showing heads when tossed. Suppose the coin is tossed repeatedly until a head occurs. Let 𝑋 denote the number of tosses required to obtain a head.
𝑋 is also called a waiting time till the first success.
𝑝 𝑋 𝑘 = 1−𝑝 𝑘−1 𝑝, 𝑘=1, 2, 3, … 𝐸 𝑋 = 1 𝑝
Notation: 𝑋∼𝐺𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐(𝑝)

26. Poisson random variable
Number of phone calls arriving at a call center per minute.
Number of jumps in a stock price in a given time interval.
Number of misprints on the front page of a newspaper.
𝑝 𝑋 𝑘 = 𝑒 −𝜆 𝜆 𝑘 𝑘! , 𝑘=0, 1, 2,… 𝐸 𝑋 =𝜆
Notation: 𝑋∼𝑃𝑜𝑖𝑠𝑠𝑜𝑛 𝜆
Can be used to approximate a 𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙 𝑛,𝑝 when 𝑛 is very large and 𝑝 is very small by setting 𝜆=𝑛𝑝.
Misprints: 𝑛 is the total number of characters and 𝑝 is the probability of a character being misprinted.

27. Uniform random variable
Let 𝑋 denote a random variable that takes any value inside [𝑎,𝑏] with equal probability.
𝑓 𝑋 𝑥 = 1 𝑏−𝑎 , 𝑖𝑓 𝑎<𝑥<𝑏 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝐹 𝑋 𝑥 = 0, 𝑖𝑓 𝑥≤𝑎 𝑥−𝑎 𝑏−𝑎 , 𝑖𝑓 𝑎<𝑥<𝑏 1, 𝑖𝑓 𝑥≥𝑏 𝐸 𝑋 = 𝑎+𝑏 𝑉𝑎𝑟 𝑋 = 𝑏−𝑎
Notation: 𝑋∼𝑈𝑛𝑖𝑓𝑜𝑟𝑚(𝑎,𝑏)

28. Exponential random variable
Time it takes for a server to complete a request.
Time it takes before your next telephone call.
𝑓 𝑋 𝑥 = 𝜆 𝑒 −𝜆𝑥 , 𝑖𝑓 𝑥≥0 0, 𝑖𝑓 𝑥< 𝐹 𝑋 𝑥 = 0, 𝑖𝑓 𝑥<0 1− 𝑒 −𝜆𝑥 , 𝑖𝑓 𝑥≥ 𝐸 𝑋 = 1 𝜆
Notation: 𝑋∼𝐸𝑥𝑝𝑜𝑛𝑒𝑛𝑡𝑖𝑎𝑙(𝜆)

29. Normal/Gaussian random variable
Velocities of molecules in a gas.
𝑓 𝑋 𝑥 = 1 𝜎 2𝜋 𝑒 − 𝑥−𝜇 𝜎 𝐸 𝑋 =𝜇 𝑉𝑎𝑟 𝑋 = 𝜎 2
Notation: 𝑋∼𝑁𝑜𝑟𝑚𝑎𝑙(𝜇, 𝜎 2 )
Arises as a limit of the sample average of independent and identically distributed (i.i.d.) random variables, e.g. 𝐵𝑖𝑛𝑜𝑚𝑖𝑎𝑙(𝑛,𝑝) as 𝑛→∞. This is known as the Central-Limit Theorem, which will be presented in a subsequent lecture.

30. Questions?