Stoicastic Method
ECE 4110: Random Signals in Communications and Signal Processing
ECE department, Cornell University, Fall 2013 Homework 2
Due September 20 at 5:00 p.m.
1. (Chebyshev Inequality) Let X1 , . . ., X be independent Geometric random variables with parameters p1 , . . . , p respectively (i.e., P (Xi = k) = pi (1 − pi )k−1 , k = 1, 2, . . .). Let random variable X to be X= i=1 Xi .
(a) Find µX = E[X]. (b) Apply the Chebyshev inequality to upper bound P (|X − µX | > a). Evaluate your upper bound for the case that p1 = · · · = p = p. What happens as → ∞? 2. (Minimum of Independent Exponential Random Variables) Assume that T1 , . . ., Tn are independent random variables and that each Ti is exponentially distributed with mean 1/µi , i = 1, . . . , n. Let T = min(T1 , . . . , Tn ). (a) Show that T is exponentially distributed. What is its mean? (b) Let the random variable K indicate the index of the Ti that is the minimum. Show that µk P (K = k) = n . i=1 µi 3. (Joint PDF and CDF) Two random variables X and Y have joint pdf: fX,Y (x, y) = csin(x + y) 0 ≤ x ≤ π/2, 0 ≤ y ≤ π/2 (a) Find the value of the constant c. (b) Find the joint cdf of X and Y . (c) Find the marginal pdf’s of X and of Y . (d) Find the mean, variance, and covariance of X and Y . 4. (Uncorrelated vs. Independent)
(a) We have seen that independence of two random variables X and Y implies that they are uncorrelated, but that the converse is not true. To see this, let Θ be uniformly distributed on [0, π] and let random variables X and Y be defined by X = cos Θ and Y = sinΘ. Show that X and Y are uncorrelated but they are not independent. (b) Now show that if we make the stronger assumption that random variables X and Y have the property that for all functions g and h, the random variables g(X) and h(Y ) are uncorrelated, then X and Y must be independent. That is, uncorrelation of the random variables is not enough, but uncorrelation of all functions of the random variables is enough to ensure
ECE department, Cornell University, Fall 2013 Homework 2
Due September 20 at 5:00 p.m.
1. (Chebyshev Inequality) Let X1 , . . ., X be independent Geometric random variables with parameters p1 , . . . , p respectively (i.e., P (Xi = k) = pi (1 − pi )k−1 , k = 1, 2, . . .). Let random variable X to be X= i=1 Xi .
(a) Find µX = E[X]. (b) Apply the Chebyshev inequality to upper bound P (|X − µX | > a). Evaluate your upper bound for the case that p1 = · · · = p = p. What happens as → ∞? 2. (Minimum of Independent Exponential Random Variables) Assume that T1 , . . ., Tn are independent random variables and that each Ti is exponentially distributed with mean 1/µi , i = 1, . . . , n. Let T = min(T1 , . . . , Tn ). (a) Show that T is exponentially distributed. What is its mean? (b) Let the random variable K indicate the index of the Ti that is the minimum. Show that µk P (K = k) = n . i=1 µi 3. (Joint PDF and CDF) Two random variables X and Y have joint pdf: fX,Y (x, y) = csin(x + y) 0 ≤ x ≤ π/2, 0 ≤ y ≤ π/2 (a) Find the value of the constant c. (b) Find the joint cdf of X and Y . (c) Find the marginal pdf’s of X and of Y . (d) Find the mean, variance, and covariance of X and Y . 4. (Uncorrelated vs. Independent)
(a) We have seen that independence of two random variables X and Y implies that they are uncorrelated, but that the converse is not true. To see this, let Θ be uniformly distributed on [0, π] and let random variables X and Y be defined by X = cos Θ and Y = sinΘ. Show that X and Y are uncorrelated but they are not independent. (b) Now show that if we make the stronger assumption that random variables X and Y have the property that for all functions g and h, the random variables g(X) and h(Y ) are uncorrelated, then X and Y must be independent. That is, uncorrelation of the random variables is not enough, but uncorrelation of all functions of the random variables is enough to ensure