Unformatted text preview: s of the random variable by x
and the associated probability by p(x)
v༇ The probabilities of all the possible values
must sum to 1, i.e. ∑ p(x ) = 1
all x value of X
probability x1
p1 x2 … p2 … xk
pk DISTRIBUTION TABLE Discrete Random Variables:
Expectation & Variance
If X has distribution
value of X
probability x1
p1 x2 p2 xk
pk …
…
k then X has expectation:
Weighted MEAN and variance
Weighted SQUARED DEVIATIONS µ = ∑ xi pi = E ( X )
i=1
k σ = ∑ ( xi − µ ) pi = Var ( X )
2 2 i=1 Why is expectation the longterm average?
“long run frequency” interpretation of probability Suppose P(event) = p. Define,
# times event occurs in n independen t trials
Fn =
n As n increases, this fraction will approach p : Fn → p CoinToss Game
v༇ Toss a coin 3 times. I will pay you 1$ for each head.
If you play the game 1000 times, how much money
do you expect to win on an average?
x 0 1 2 3 Probability 1/8 3/8 3/8 1/8 v༇ In each game, denote the following events as:
q༇ A={no head}; B={1 head}; C={2 heads}; D={3 heads}. Why is the expectation the longterm average?
v༇ Imagine playing the game n times, with n large.
Total “sample” winnings: X1 + X 2 + + X n = 0( #A) + 1( # B) + 2( # C)+3 (# D)
Mean of sample winnings:
!# A$
! # B$
! # C$
! # D$
Xn = 0 #
& + 1#
& + 2#
& + 3#
&
"n%
"n%
"n%
"n%
1/8 3/8 3/8 1/8 X n → 0(1 / 8) + 1(3 / 8) + 2(3 / 8) + 3(1 / 8) = E ( X ) Law of large numbers
As we do many independent repetitions of the
experiment, drawing more and more numbers from
the same distribution, the mean of our sample will
approach the mean of the distribution more and
more closely:
X n → E( X ) as n increases Example: Calculation of µ & SD
v༇ Suppose a day trader buys one share of IBM.
v༇ Let X represent the change in price of IBM.
v༇ She pays $100 today, and the price tomorrow can be
either $110, $100 or $90.
Stock Price Increases Remains Same Decrease Change (x) Probability $10 0.2 0 0.7  $10 0.1 Example: Calculation of µ & SD Stock Price Increases Remains Same Decrease Change (x) $10 0  $10 Probability 0.2 0.7 0.1 v༇ Find µ: The day trader expects on average to make _____$ on every share of IBM she buys. v༇ Find SD: How risky is this investment? Deviation
Change Probability
(x)
10 0.2 0 0.7 10 0.1 xi − µ i Squared Deviation ( xi − µ i ) 2 Expected value and SD: properties
Adding or Subtracting a Constant (c)
• Changes the expected value by a fixed amount:
E(X ± c) = E(X) ± c
• Does not change the standard deviation:
SD(X ± c) = SD(X) Expected value and SD: properties
v༇ Multiplying by a Constant (c)
q༇ Changes the expected by the same scale
E(cX) = c E(X). q༇ Changes th...
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 Fall '07
 Lv
 Probability, Probability theory

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