Worksheets(12-13)

# Worksheets(12-13) - #12 22S:8(Random Variables 4.3(Russo...

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#12 22S:8 (Random Variables – 4.3 ) (Russo) (1) Let X be the result of one spin of the spinner shown. 22 What is X ? Answer: We don't know until we do the experiment. X is called a random variable . We can list the possible values of X and the corresponding probabilities. Possibility 1 2 5 draw the probability histogram Probability 1/2 1/3 1/6 pdf of X: f(x) = 1/(x+1) x = 1, 2, 5 ___________________________________ 1 2 5 probability density function E(X) = µ = sum of possibilities × probabilities = 1(1/2) + 2(1/3) + 5(1/6) = 2 E(X 2 ) = Var(X) = E[ (X - µ) 2 ] = SD(X) = sqrt Var(X) = Important computational formula : Var(X) = E(X 2 ) - µ 2 = (2) Suppose the random varaible Y has pdf f(y) = y 2 /55 y = 1, 2, 3, 4, 5 E(Y) = ___________________________________ 1 2 3 4 5 Var(Y) = P(Y > 2) = P(Y is odd valued) = P(Y 2 + 3 < 12) = P(Y = 1 or Y = 2) = P( Y < 4 | Y > 1 ) =

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#12a (3) Suppose the random variable Q has pdf f(q) = C(q+1) 1/2 q = 0, 3, 8, 24 Find the value of C. __________________________________ 0 3 8 24 E(Q) = probability histogram Var(Q) = P(5Q + 3 is even valued) = (4) Suppose T = 3X + 5 where X is the random variable from problem (1). What are the possible values of T ? Find the pdf of T. E(T) = How are E(T) and E(X) related ? Var(T) = How are Var(T) and Var(X) related ? How are SD(T) and SD(X) related ? IMPORTANT RULES E(aX + b) = aE(X) + b Var(aX + b) = a 2 Var(X) E(X + Y) = E(X) + E(Y) Var(X + Y) = Var(X) + Var(Y) if X and Y are independent
#12b (application of the rules for expectation & variance) Suppose X, Y & Z are independent random variables (representing the value of a single share of stock in three different companies one year from today), and that E(X) = 12 E(Y) = 8 E(Z) = 10 Var(X) = 5 Var(Y) = 3 Var(Z) = 2 E(8X) = E(5X + 3Y - 2Z -1) = E(X 2 ) = E(5X 2 ) = E(5X 2 - 8) = E(5X 2 - 2Y + 3Z 2 + 4) = Var(5Y) = Var(4Y + 8) = Var(-Z) = Suppose X, Y & Z are independent.

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