10 mean variance cheb

10 mean variance cheb - The Mean and Variance of a Random...

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3/26/10 The Mean and Variance of a Random Variable Two important quantities that describe the behavior of a random variable and PMF. The mean is also equivalently known as the Expected Value. We can denote the mean/expected value or expectation of a random variable X using E[X] or ˚X or often just }
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3/26/10 Example 1 Two loaded dice (X, and Y) have the X 1 2 3 4 5 6 PX(x) 0.01 0.05 0.1 0.2 0.3 0.34 PY(x) 0.34 0.3 0.2 0.1 0.05 0.01
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3/26/10 Example Find the mean of the Bernoulli Distribution. p Note that when a PMF is written in generic form with parameters, the mean is a function of the parameters Find the mean of the Geometric
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3/26/10 Important Properties of Suppose X is a random variable, and a and b are constants. Suppose we define Y as follows: Y = aX + b Then Y is a random variable, but with a Suppose X was the outcome of a die, and y 12 14 16 18 20 22 PY(y) 1/6 1/6 1/6 1/6 1/6 1/6
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3/26/10 Property 1 Suppose we create a new random variable Y from X as Y = aX+b. Given the mean of X is E(X), what is the mean of Y? E[Y] = E[aX+b] = aE[X]+b Proof: PY(y) = P(Y=y) = P(Y=ax+b) = PX(x)
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3/26/10 Property 2 Let g() be any function, and X be a random variable. Let Y = g(X), thus Y is a random variable as well. Then: E[g(X)] = & g(x) PX(x) However the following is NOT CORRECT E[ g(X)] = > ] )
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3/26/10 Example The PMF of X is : P(X=1) = 0.3 P(X=2) = 0.7. Let Y = X2. a) Calculate the PMF of Y. a) Calculate E[Y]
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3/26/10 Property 3 Suppose we have two random variables X and Y.
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10 mean variance cheb - The Mean and Variance of a Random...

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