Total sample winnings x1 x 2 x n 0 a 1 b

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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 long-term 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 Coin-Toss 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 long-term 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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