1. Expectation and Variance
X: sum of two die rolls
X = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
P(X=2) = 1/36
P(X=3) = 1/18
P(X=4) = 1/12
P(X=5) = 1/9
P(X=7) = 1/6
P(X=13) = ?
P(X=13) = 0
𝐸 𝑋 =(2)𝑃 𝑋=2 +(3)𝑃 𝑋=3 +(4)𝑃 𝑋=4 +⋯+ 12 𝑃(𝑋=12) =
= = = 42 6 =7 X
P(X=x) 2
1/36 3
1/18 4
1/12 5
1/9 6
5/36 7
1/6 X
P(X=x) 8
5/36 9
1/9 10
1/12 11
1/18 12
1/36

2. Expectation and Variance
X: sum of two die rolls
X = 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
P(X=2) = 1/36
P(X=3) = 1/18
P(X=4) = 1/12
P(X=5) = 1/9
P(X=7) = 1/6
𝑉𝑎𝑟 𝑋 =𝐸 𝑋 2 − 𝐸 𝑋 2
𝐸 𝑋 2 = (2) 2 𝑃 𝑋=2 + (3) 2 𝑃 𝑋=3 + (4) 2 𝑃 𝑋=4 +⋯+ (12) 2 𝑃 𝑋=12
= (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)
= = = ≈
𝑉𝑎𝑟 𝑋 =𝐸 𝑋 2 − 𝐸 𝑋 2 = − = − = = 35 6 ≈5.8333
𝑆𝐷 𝑋 = =2.4152 X
P(X=x) 2
1/36 3
1/18 4
1/12 5
1/9 6
5/36 7
1/6 X
P(X=x) 8
5/36 9
1/9 10
1/12 11
1/18 12
1/36

3. Expectation and Variance
Suppose we have multiple Discrete Random Variables 𝑋 1 , 𝑋 2 ,…, 𝑋 𝐾 with all random variables mutually independent. Suppose we create a new random variable by taking a weighted sum of the others:
𝑌=𝑐+ 𝑘=1 𝐾 𝑏 𝑘 𝑋 𝑘 =𝑐+ 𝑏 1 𝑋 1 + 𝑏 2 𝑋 2 +⋯+ 𝑏 𝐾 𝑋 𝐾 .
Expected value of 𝑌:
𝐸 𝑌 =𝐸 𝑐+ 𝑘=1 𝐾 𝑏 𝑘 𝑋 𝑘 =𝐸 𝑐 + 𝑘=1 𝐾 𝐸( 𝑏 𝑘 𝑋 𝑘 ) =𝑐+ 𝑘=1 𝐾 𝑏 𝑘 𝐸( 𝑋 𝑘 )
=𝑐+ 𝑏 1 𝐸 𝑋 1 + 𝑏 2 𝐸 𝑋 2 +⋯+ 𝑏 𝐾 𝐸 𝑋 𝐾 =𝑐+ 𝑏 1 𝜇 𝑋 1 + 𝑏 2 𝜇 𝑋 2 +⋯+ 𝑏 𝐾 𝜇 𝑋 𝐾 = 𝜇 𝑌
Variance of 𝑌:
𝑉𝑎𝑟 𝑌 = Var 𝑐+ 𝑘=1 𝐾 𝑏 𝑘 𝑋 𝑘 = 𝑘=1 𝐾 𝑉𝑎𝑟 𝑏 𝑘 𝑋 𝑘 = 𝑘=1 𝐾 𝑏 𝑘 2 𝑉𝑎𝑟( 𝑋 𝑘 )
= 𝑏 1 2 𝑉𝑎𝑟 𝑋 1 + 𝑏 2 2 𝑉𝑎𝑟 𝑋 2 +⋯+ 𝑏 𝐾 2 𝑉𝑎𝑟 𝑋 𝐾
= 𝑏 1 2 𝜎 𝑋 𝑏 2 2 𝜎 𝑋 ⋯+ 𝑏 𝐾 2 𝜎 𝑋 𝐾 2 = 𝜎 𝑌 2
The standard deviation of Y is just the square root of 𝜎 𝑌 2 .

4. Expectation and Variance
𝑋 𝑖 : face value of a fair die roll on the ith roll.
𝑋 𝑖 = 1, 2, 3, 4, 5, 6
𝑃 𝑋 𝑖 =1 =1/6
𝑃 𝑋 𝑖 =2 =1/6
𝑃 𝑋 𝑖 =3 =1/6
𝑃 𝑋 𝑖 =4 =1/6
𝑃 𝑋 𝑖 =5 =1/6
𝑃 𝑋 𝑖 =6 =1/6
We know that 𝐸 𝑋 𝑖 =3.5, and that 𝑉𝑎𝑟 𝑋 𝑖 =
Now, suppose that we want the expected value and variance of a new random variable Y, defined as the sum of two independently thrown dice. We find the following:
𝐸 𝑌 =𝐸 𝑋 1 + 𝑋 2 =𝐸 𝑋 1 +𝐸 𝑋 2 = =7
𝑉𝑎𝑟 𝑌 =𝑉𝑎𝑟 𝑋 1 + 𝑋 2 =𝑉𝑎𝑟 𝑋 1 +𝑉𝑎𝑟 𝑋 2 = = = 35 6 =5.8333
This is exactly the same as we found by computed the expected value and variance of the sum of two dice directly!
𝑋 𝑖
𝑷( 𝑋 𝑖 = 𝑥 𝑖 ) 1
1/6 2 3 4 5 6

5. Continuous Random Variable
Consider a random variable X with values from 0 to 1
Discrete
Continuous
∆𝑥=0.1?
∆𝑥=0.01?
∆𝑥=0.001? X
P(X=x)
0.4 1
0.6 X
P(X=x)
0.1
0.2 ⋮
0.9
1.0 X
P(X=x)
0.01
0.02 ⋮
0.99
1.00 X
P(X=x)
0.001
0.002 ⋮
0.999
1.000
What increment do you use???

6. Continuous Random Variable
Consider a random variable X with values from 0 to 1
Discrete
Continuous
∆𝑥=0.1?
∆𝑥=0.01?
∆𝑥=0.001? X
P(X=x)
0.4 1
0.6 X
P(X=x)
0.1
0.2 ⋮
0.9
1.0 X
P(X=x)
0.01
0.02 ⋮
0.99
1.00 X
P(X=x)
0.001
0.002 ⋮
0.999
1.000
What increment do you use???

7. Continuous Random Variable
Every continuous random variable is defined by a probability density function (pdf)
Probability density function for continuous random variable 𝑋: 𝑓(𝑥)
Probability of 𝑋 less than or equal to some value 𝑎, 𝑃 𝑋≤𝑎 = 𝐹 𝑋 (𝑎), is defined as the area under the curve to the left of 𝑎
𝑃 𝑋≤𝑎 = −∞ 𝑎 𝑓 𝑥 𝑑𝑥
In this example, values of
𝑋 range from −∞ to ∞ a

8. Continuous Random Variable
Every continuous random variable is defined by a probability density function (pdf)
Probability density function for continuous random variable 𝑋: 𝑓(𝑥)
Probability of 𝑋 less than or equal to some value 𝑎, 𝑃 𝑋≤𝑎 = 𝐹 𝑋 (𝑎), is defined as the area under the curve to the left of 𝑎
𝑃 𝑋≤𝑎 = −∞ 𝑎 𝑓 𝑥 𝑑𝑥
Probability of 𝑋 greater than or equal to some value 𝑎, 𝑃 𝑋≥𝑎 , is defined as the area under the curve to the right of 𝑎
𝑃 𝑋≥𝑎 = 𝑎 ∞ 𝑓 𝑥 𝑑𝑥
In this example, values of
𝑋 range from −∞ to ∞ a

9. Continuous Random Variable
Every continuous random variable is defined by a probability density function (pdf)
Probability density function for continuous random variable 𝑋: 𝑓(𝑥)
Probability of 𝑋 less than or equal to some value 𝑎, 𝑃 𝑋≤𝑎 = 𝐹 𝑋 (𝑎), is defined as the area under the curve to the left of 𝑎
𝑃 𝑋≤𝑎 = −∞ 𝑎 𝑓 𝑥 𝑑𝑥
Probability of 𝑋 greater than or equal to some value 𝑎, 𝑃 𝑋≥𝑎 , is defined as the area under the curve to the right of 𝑎
𝑃 𝑋≥𝑎 = 𝑎 ∞ 𝑓 𝑥 𝑑𝑥
Probability of 𝑋 between 𝑎 and 𝑏, 𝑃 𝑎≤𝑋≤𝑏 , is defined as the area under the curve between 𝑎 and 𝑏
𝑃 𝑎≤𝑋≤𝑏 = 𝑎 𝑏 𝑓 𝑥 𝑑𝑥
In this example, values of
𝑋 range from −∞ to ∞ a a b

10. Continuous Random Variable
Every continuous random variable is defined by a probability density function (pdf)
Probability density function for continuous random variable 𝑋: 𝑓(𝑥)
Probability of the entire range of 𝑋, 𝑃 −∞≤𝑋≤∞ , is the area under the entire curve
𝑃 −∞≤𝑋≤∞ = −∞ ∞ 𝑓 𝑥 𝑑𝑥 =1  Probability of all possible values of 𝑋
In this example, values of
𝑋 range from −∞ to ∞

11. Continuous Random Variable
Every continuous random variable is defined by a probability density function (pdf)
Probability density function for continuous random variable 𝑋: 𝑓(𝑥)
Probability of the entire range of 𝑋, 𝑃 −∞≤𝑋≤∞ , is the area under the entire curve
𝑃 −∞≤𝑋≤∞ = −∞ ∞ 𝑓 𝑥 𝑑𝑥 =1  Probability of all possible values of 𝑋
Probability of 𝑋 between 𝑎 and 𝑎, 𝑃 𝑎≤𝑋≤𝑎 =𝑃 𝑋=𝑎 , is the area under the curve between 𝑎 and 𝑎
𝑃 𝑋=𝑎 = 𝑎 𝑎 𝑓 𝑥 𝑑𝑥 =0  Area doesn’t exist! Area of a line is 0.
A continuous random variable theoretically has an infinite number of possible values
The probability of 𝑋 equal to an exact point value out of an infinite number of choices = 1 ∞ =0
In this example, values of
𝑋 range from −∞ to ∞ a

12. Continuous Random Variable
For continuous 𝑋, since 𝑃 𝑋=𝑎 =0, ≤ and ≥ are interchangeable with < and >, respectively
𝑃 𝑋≤𝑎 =𝑃 𝑋<𝑎 +𝑃 𝑋=𝑎 =𝑃 𝑋<𝑎 +0=𝑃 𝑋<𝑎
𝑃 𝑋≥𝑎 =𝑃 𝑋>𝑎 +𝑃 𝑋=𝑎 =𝑃 𝑋>𝑎 +0=𝑃(𝑋>𝑎)
𝑃 𝑎≤𝑋≤𝑏 =𝑃 𝑎≤𝑋<𝑏 =𝑃 𝑎<𝑋≤𝑏 =𝑃(𝑎<𝑋<𝑏)
Caution: not true for discrete 𝑋 because 𝑃 𝑋=𝑎 is not 0 (given that 𝑎 is a possible value of 𝑋)

13. Continuous Random Variable
Complement rule
𝑃 𝑋≤𝑎 =1−𝑃 𝑋>𝑎 (𝑋>𝑎 and 𝑋≤𝑎 are complements)
𝑃(𝑋≤𝑎)
𝑃 −∞≤𝑋≤∞ =1
𝑃(𝑋>𝑎) = − a a

14. Continuous Random Variable
Interval
 𝑃 𝑎≤𝑋≤𝑏 =𝑃 𝑋≥𝑎 −𝑃(𝑋≥𝑏)
𝑃 𝑎≤𝑋≤𝑏
𝑃(𝑋≥𝑎)
𝑃(𝑋≥𝑏) = − a b a b