View
5
Download
0
Category
Preview:
Citation preview
Feb 27: Expectation, Variance, and Standard
Deviation
In-class Midterm Exam MOVED
to 3/10
Goals for today
What are mean, variance, and standard deviation?
What is the difference between distribution mean/variance and sample mean/variance?
When are mean and variance informative, and when are they misleading?
What is the 68/95/99.7 rule?
Mean is a balance pointtorque = force × distance
Mean is a balance pointtorque = force × distance
Mean is a balance pointtorque = force × distance
Mean is a balance pointtorque = force × distance
balance point is where we get equal torque on both sides
Mean is a balance pointtorque = force × distance
balance point is where we get equal torque on both sides 5
5
6666
777 10
6.44
Mean is a balance pointtorque = force × distance
balance point is where we get equal torque on both sides 5
5
6666
777 10
x - μ =
6 - 6.44 = -0.44
Mean is a balance pointtorque = force × distance
balance point is where we get equal torque on both sides 5
5
6666
777 10
4 × -0.44
Mean is a balance pointtorque = force × distance
balance point is where we get equal torque on both sides
Σ (x - μ) = 0
Σ x = Nμ
(Σ x)/N = μ
55
6666
777 10
mean = average
Mean is a balance pointtorque = force × distance
balance point is where we get equal torque on both sides 5
5
6666
777 10
6.44
Mean is a balance pointtorque = force × distance
balance point is where we get equal torque on both sides 5
5
6666
777 8
Mean is a balance pointtorque = force × distance
balance point is where we get equal torque on both sides 5
5
6666
777 8
Mean is sensitive to outliers
55
6666
777 17
5 5 6 6 6 6 7 7 10
Median ignores values
5 5 6 6 6 6 7 7 10
Median ignores values
5 5 6 6 6 6 7 7 328
Median ignores values
The sum of squared distances to the meanx = [2, 3, 7]
2 3 7
2 3 7
The sum of squared distances to the mean
2 3 7
The sum of squared distances to the mean
2x2
1x1
3x3
2 3 7Σ (x - μ)2
N
= (4 + 1 + 9)/3 = 4.66
Variance: mean squared distances to the mean
2 3 7Σ (x - μ)2
N
= (4 + 1 + 9)/3 = 4.66
Variance: mean squared distances to the mean
2.16x2.16
2 3 7Σ (x - μ)2
N
= (4 + 1 + 9)/3 = 4.66
Variance: mean squared distances to the mean
2.16
2 3 7
Standard deviation: square root of mean squared distances to the mean
2.64x2.64
2 3 7Σ (x - μ)2
N-1
= (4 + 1 + 9)/2 = 7
Variance: alternative form
2x2
1x1
3x3
2 3 7
Mean is the point that minimizes variance for a fixed data set
d/dμ Σ (x - μ)2
= 2 Σ (x - μ)
Σ (x - μ) = 0
Goals for today
What are mean, variance, and standard deviation?
What is the difference between distribution mean/variance and sample mean/variance?
When are mean and variance informative, and when are they misleading?
What is the 68/95/99.7 rule?
Mean is a balance point for a distributiontorque = force × distance
balance point is where we get equal torque on both sides
P(2)
P(3)
P(4)
P(10)
Mean is a balance point for a distributiontorque = force × distance
balance point is where we get equal torque on both sides
μ = Σ x P(x)
P(2)
P(3)
P(4)
P(10)
mean = average = expectation
What are the expectations of these two dice?
P(6)=1/2
P(6)=1/6
μ = E[x]
= Σ x P(x)
What are the expectations of these two dice?
P(6)=1/2
P(6)=1/6
μ = E[x]
= Σ x P(x)
"expectation of x"
What are the expectations of these two dice?
P(6)=1/6 E[x] = Σ xP(x)= 1×.16 + 2×.16 + ... + 6×.16= (1 + 2 + ... + 6) × .16= 21 / 6 = 3.5
What are the expectations of these two dice?
P(6)=1/6
μ = E[x]
= Σ x P(x)
= Σ x / Nonly if P(x) is uniform for all x
What are the expectations of these two dice?
P(6)=1/2
E[x] = Σ xP(x)= 1×.1 + 2×.1 + ... + 6×.5= .1 × (1 + 2 + ... + 5) + 3= 1.5 + 3 = 4.5
What are the variances of these two dice?
P(6)=1/2
P(6)=1/6
σ2 = E[Σ (x-μ)2]
= Σ (x-μ)2 P(x)
Which has greater variance?
P(6)=1/2
P(6)=1/6
Variance of uniform distribution
P(6)=1/6 var[x] = Σ (x-μ)2 P(x)= (1-3.5)2×.16 + ... +
(6-3.5)2×.16= -2.52×.16 + -1.52×.16 + ... +
2.52×.16= 2.916
Variance of non-uniform distribution
P(6)=1/2
var[x] = Σ (x-μ)2 P(x)= (1-4.5)2×.1 + ... + (6-4.5)2×.5= -3.52×.1 + ... + 1.52×.5= 3.25
Which has greater variance?
P(6)=1/2
P(6)=1/6
Sample mean/var vs. Distribution mean/var
sample distribution
mean x̄ = Σ x/N μ = Σ x P(x)
variance s2 = Σ(x-x̄)2/N σ2 = Σ (x-μ)2 P(x)
Sample mean/var vs. Distribution mean/var
sample distribution
mean x̄ = Σ x/N μ = Σ x P(x)
variance s2 = Σ(x-x̄)2/N σ2 = Σ (x-μ)2 P(x)
Distribution vs. Sample with dice
Mean and variance for distributions
mean variance
binomial np np(1-p)
geometric 1/p (1-p)/p2
Poisson λ λ
Distribution vs. Samplewith parametric distributions
Goals for today
What are mean, variance, and standard deviation?
What is the difference between distribution mean/variance and sample mean/variance?
When are mean and variance informative, and when are they misleading?
What is the 68/95/99.7 rule?
Recommended