Lec12 Mean & SD of h(x)

Lec12 Mean & SD of h(x) - r.v. C = 5+2X = cost to...

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Mean & Std Deviation of a Function of a Discrete random variable 1
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Example: What is a function of a random variable? 2 r.v. X= number of defects on a part. Here is f(x): The rework cost is $2 per defect r.v. C = rework cost r.v. C(X)=2X x 0 1 2 f(x) .3 .5 .2 Function of r.v. X
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More… cost is function of r.v. X = #defects 3 Another possible cost function: If there are no defects, then there is no rework cost. However if there are one or two defects, then there is a fixed cost of $5 plus $2 per defect. Define r.v. C Write C(X)= Find E(C) x 0 1 2 f(x) .3 .5 .2 C
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Mean of a Function of a Random Variable 4
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Use what you know to find 5 x 0 1 2 f(x) .3 .5 .2 C C2 C2 is a function of r.v. C We know how to get E(C2 )
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Example: Getting from the fire house to a fire 6 The distance to a fire is 1 or 2 miles with probability 0.7 or 0.3. The cost to travel to the fire is 5 + 2X, and the time to travel is X2. Find the mean of the distance, cost, and time to get from the fire house to a fire. r.v. X = distance to fire
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Unformatted text preview: r.v. C = 5+2X = cost to travel X miles r.v. T = X2 = time to travel X miles Find mean of the distance, cost, and time to get from fire house to fire 7 f(x) = P(X=x) .7 .3 X, distance 1 2 C=5+2X, cost ? ? T=X2, time ? ? MoreNow find the standard deviation of cost & time to get to fire 8 f(x) = P(X=x) 0.7 0.3 X, distance 1 2 C=5+2X, cost C2 T T2 True: E(5+2X)=5+2E(X) False: E(5+X3)=5+[E(X)]3 9 Which of these functions is a linear function of X? 10 Given r.v. X, f(0)=.2 f(1)=.5 f(2)=.3 E(X)=1.1 V(X)=.49 h(X)=4X h(X)=X2 +2X+1 Find mean and variance of three h(X) 11 Given r.v. X, f(0)=.2 f(1)=.5 f(2)=.3 E(X)=1.1 V(X)=.49 h(X)=4X h(X)=X2 +2X+1 Proof: Mean and Variance of a Linear Function of r.v. X 12 Given r.v. X with possible values xi, i=1,2, . .. and a linear function of X, h(X) = a+bX E(h(X)) = a + b E(X) V(h(X)) = b2 V(X) More 13...
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Lec12 Mean & SD of h(x) - r.v. C = 5+2X = cost to...

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