MIT6_042JF10_rec22

MIT6_042JF10_rec22 - j s are mutually independent and take...

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6.042/18.062J Mathematics for Computer Science December 3, 2010 Tom Leighton and Marten van Dijk Problems for Recitation 22 1 Properties of Variance In this problem we will study some properties of the variance and the standard deviation of random variables. a. Show that for any random variable R , Var [ R ] = E [ R 2 ] E 2 [ R ]. b. Show that for any random variable R and constants a and b , Var [ aR + b ] = a 2 Var [ R ] . Conclude that the standard deviation of aR + b is a times the standard deviation of R . c. Show that if R 1 and R 2 are independent random variables, then Var [ R 1 + R 2 ] = Var [ R 1 ] + Var [ R 2 ] . d. Give an example of random variables R 1 and R 2 for which Var [ R 1 + R 2 ] = Var [ R 1 ] + Var [ R 2 ] . e. Compute the variance and standard deviation of the Binomial distribution H n,p with parameters n and p . f. Let’s say we have a random variable T such that T = j n =1 T j , where all of the T
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Unformatted text preview: j s are mutually independent and take values in the range [0 , 1]. Prove that Var(T) Ex(T). Well use this result in lecture tomorrow. Hint: Upper bound Var [ T j ] with E [ T j ] using the defnition oF variance in part (a) and the rule For computing the expectation oF a Function oF a random variable. MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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MIT6_042JF10_rec22 - j s are mutually independent and take...

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