Variance Variance of a random variable Standard deviation
Variance • Variance of a random variable • Standard deviation
Expected Value • X takes values at random k P(X=k) 1. 2 2. 6 3. 2
Measuring the Spread • • Variance = how spread out the pdf is Randomness ? ? Size ? ? Variance = average squared distance from the mean
Distance from the Mean E(X)
Notation • – μ Greek letter mu (for mean) – Population mean • – σ little sigma (for standard deviation) – Population variance
Average Squared Distance from the Mean • • • Mean = E(X) Call it E(X) Distance from the Mean = X – μ Squared Distance = Average = An equivalent formula
Another Expectation • E(X 2) = ∑ k 2 P(X=k) k 2 k P(X=k) 1 1. 2 4 2. 6 E(X 2) = 1 (. 2) + 4(. 6) + 9(. 2) =. 2 + 2. 4 + 1. 8 = 4. 4 9 3. 2
Variance • • • Var(X) = E(X 2) – [E(X)]2 E(X) = 2 [E(X)]2 =4 E(X 2) = 4. 4 Var(X) = 4. 4 – 4 = 0. 4 Standard deviation of X = 0. 632
Other way of computing • Var(X) = E(X – μ)2 Expected squared distance from mean (k – μ)2 k –μ k 1 -1 1 0 0 2 1 1 3 P(X=k) . 2 . 6 . 2 E(X – μ)2 = (1 – 2)2(. 2) + (2 – 2)2(. 6) + (3 – 2)2(. 2) = 1(. 2) + (0)(. 6) + 1(. 2) =. 4
Variance • • • Var(X) = E(X 2) – µ 2 E(X) = 2 µ 2 =4 E(X 2) = 4. 4 Var(X) = 4. 4 – 4 = 0. 4 Standard deviation of X = 0. 632
E(X 2) is not [E(X)]2 • Square then average = E(X 2) • Average then square = µ 2 • E(X 2) ≥ [E(X)]2 Equal only in trivial cases Variance = Difference E(X 2) - [E(X)]2 ≥ 0
Variance = 0
Example X -2 0 1 P . 25
Mirror Image X -1 0 2 P . 25
Var(X) = 1
Var(X) = 4 !!
Standard Deviation • Variance in squared units Var(Height) = inches 2 • •
Example k -2 0 1 P(X=k) . 1 . 5 . 4 E(X) = -. 2 + 0 +. 4 =. 2 E(X 2) = 4(. 1) + 0 + 1(. 4) =. 8 Var(X) =. 8 – 0. 04 =. 76
Variance of the Bet
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Flipping Coins • Flip a coin 4 times – X = number of heads – P(X=4) = (½) (½) = 1/16 – P(X=0) = P( 4 Tails) = 1/16 – P(X=1) = ? ? ?
Sample Space HHHH HTHH THHH TTHH HHHT HTHT THHT TTHT HHTH HTTH THTH TTTH HHTT HTTT k P(X=k) THTT TTTT 0 1 2 3 4 1/16 1/4 3/8 1/4 1/16
Two Games 1. You win $1000. E(X) = $1000 2. You win $10, 000 with probability 1/10 or $0. 01 with probability 9/10. E(X) = $1000. 009 Which would you play?
- Slides: 25