ECE 450 1 Homework Set 3 0. Consider the random variables X and Y, whose values are a function of the number showing when a single die is tossed, as show below: Exp. Outcome 1 3 4 5 6 X 3 3 4 4 Y 0 1 3 4 5 a. Calculate: F XY (3, 4); F XY (1, 3); and F XY (5, 5). b. Sketch: f XY (x, y). Hint: Since these are discrete RV s, the graph should consist of functions, located at several points in the x-y plane. 1. Given the joint density function f XY (x,y) = 0.5 1<x<, <y<3 0.5 <x<4, <y<3 a. Find E(XY). b. Find f X (x Y =.5). c. Find Pr(X < 1.5).. Let X, Y, and Z be independent random variables with respective means 10, 0, and 30 and respective variances 0, 4 and 9. Let U = XY and let V=YZ. a. Find cov(u,v). b. Find Pr(X = 10). 3. Let X and Y be Gaussian independent random variables with mean 0 and variance 4. Let Z = X +Y. Find a. an expression for the probability density function for Z. b. Pr( Z < 1) Hint: you should recognize this Z as a particular kind of RV. 4. Let X 1 and X be two random variables with correlation coefficient r =.5; suppose the variance of X 1 is 4 and the variance of X is 1. a. Write out the terms in the two-by-two covariance matrix, C = C ij. b. If both random variables have mean 10, find the correlation, E(X 1 X ). 5. Suppose two random variables X and Y have the same covariance as they have correlation; i.e., cov(x,y) = E(XY). What can you conclude about these two random variables? 6. Suppose for two random variables X and Y, E(XY) = E(X) E(Y).
ECE 450 What can you conclude about these two random variables? 7. Consider the joint probability density function for random variables X and Y that is uniform on a circle of radius 10 in the X-Y plane; i.e., f XY (x,y) = k x +y < 100 a. Find k. b. Find Pr(X + Y < 5). 8. Let X be a discrete random variable taking on the two values ±10 with equal probability. Let Y be a uniform random variable on the interval (-1,1). If Z = X + Y, and X and Y are independent, find the probability density function for the random variable Z. 9. Consider the joint pdf f(x, y) = a. Find the marginal pdf s, f X (x) and f Y (y). b. Find the conditional pdf, f X (x y). c. Find Pr(X < 1, Y < ). 1 x 9y exp 18 u(y) 18 10. Random variables X and Y have joint pdf a. Find the marginal density, f Y (y). b. Find the conditional pdf, f X (x y). f(x, y) = ½, 0 < x < y, 0 < y < 11. (Yates & Goodman) Random variables X and Y have joint pdf fxy(x,y) = 4 x y 0 x 1, 0 y 1 a. Find E(X) and var(x). b. Find E(Y) and var(y). c. Find cov(x, Y). d. Find E(X + Y). e. Find var(x + Y). 1. Two random variables X and Y have the joint probability density function
ECE 450 3 f XY (x,y) = 1 30 1 40 0 < X < 4, 0 < Y < 3 4 < X < 10, 0 < Y < 4 () a. Find Pr[ (3 < X < 5) ( < Y < 4) ] b. Find Pr[ X < Y < ] 13. Two random variables have mean zero and variances 16 and 36. Find the variance of their sum if: a. their correlation coefficient is 0; b. their correlation coefficient is 0.5; c. their correlation coefficient is -0.5. 14. Random variable X has variance 9, and statistically independent variable Y has variance 16. Say Z = X + Y. Find a. the correlation coefficient for X and Z; b. the correlation coefficient for Y and Z; c. the variance of Z. 15. The resistance values of a supply of resistors are independent RV s, uniformly distributed between 100 and 10 ohms. If resistors are selected at random and connected in series, find a. the pdf for the RV representing the series combination of the resistors; b. the probability that the series combination of the resistors is at least 00. 16a. Fill in the blanks for the covariance matrix for random vector X, as shown below: 9.8 4 1 3 3 5.8 3 10 16 16b. Find cov(x 1, X ). 16c. Find var(x 4 ). 17. (Yates & Goodman) Temperature measurements are taken 3 times a day, resulting in Gaussian RV s X 1 (at 6 AM), X (at 1 noon), and X 3 (a 6 PM), each with variance 16 (deg. ), and with respective means 50, 6, and 58. The covariance matrix for the measurement vector is
ECE 450 4 C X = 16 1.8 11. 1.8 16 1.8 11. 1.8 16 a. Write the joint pdf of X 1 and X in vector form. b. Write the joint pdf of X 1 and X in algebraic form. c. Write the joint pdf of X 1, X, and X 3 in vector form. 18. Consider the example given in Lecture 9, Part 1, pp. -4. Verify the probabilities: Pr(B = 0, G = ) =.0875 Pr(B = 3, G = 0) =.0375 using arguments similar to those given on p. 3. 19. Verify that if = 0 in the equation for the pdf of jointly Gaussian RV s X and Y given on p. 6 of Lecture 9, Part 1, then the RV s X and Y are independent. (Of course you need to show that the joint pdf factors into two separate functions, one of which is a Gaussian pdf about the RV X, and the other of which is a Gaussian pdf about the RV Y.) 0a. Use MATLAB s meshgrid function to plot the joint pdf of Gaussian RV s, X and Y, given the parameters: m x = 0 = m y ; x = 1 = y ; =.95 0b. Repeat part a, but change the parameter to =.. 0c. Repeat part a, but change the parameter to =.0. 0d. Plot all 3 graph on a single page, using MATLAB s subplot function. Hint: It s OK to use the code given in Lecture 9, Part 1. 1. Given the Gaussian RV s X and Y with parameters as described in problem number 0c, find F XY (1, 3) = Pr(X < 1, Y < 3). (You may use table or MATLAB s functions: qfunc or normcdf.. Suppose that received signal Y is given by Y = X + N, where X is the transmitted waveform value and N is noise. a. Find f Y (y x) if the noise N is Gaussian, N(0, = ), and the given value of X is X = 3. b. Find f Y (y x) if N is uniform on the interval (, ) and the given value of X is X = 10.
ECE 450 5 3. Consider a random vector X consisting of 3 independent random variables X 1, X and X 3, each of which is Gaussian with = 4, assuming that the means are 1, and 3 (respectively). Find: a. the correlation matrix, R X ; and b. the covariance matrix, C X. Random Process Questions (Lecture 10 p. 5 end of Lecture 11) 4. A time function is generated by flipping coins once every second, and assigning a value of +1 to heads and -1 to tails. The time function takes a constant value equal to the sum of the values generated by the coin toss for one second, and then changes value at the time of the next coin toss. a. Starting at time 0, suppose the coin toss values are as follows: time coin #1 coin # 0 H T 1 T T H T 3 H H 4 T H... Graph the sample function from time t = 0 to t = 5 sec., corresponding to this experimental outcome. b. How many possible sample functions are there for this random process of duration 5 seconds? 5. (Yates & Goodwin) A WSS process X(t) with autocorrelation function R X (t) = exp(- 4 ) is input to a filter with transfer function H(f) = 1 0 f, a. Find the total average power of the input X(t). b. Find the output power spectral density, S Y (f). c. Find the total average power of the output process Y(t).
ECE 450 6 6. A WSS random process has autocorrelation function R X () = 9 exp(-4 ) + 16 a. Find the mean value, mean-squared value and the variance of this random process. b. Find the power spectral density, S X (w), for this RP. 7. Find the total power in the WSS random process with power spectral density S X (f) = 10, 4 < f < 5 (Note: This is an example of band-limited white noise; i.e., noise with a flat spectrum over a limited range of frequencies (and where).) 8a. Find the autocorrelation function and the mean for a WSS random process with power spectral density: 4 S X () =. 16 8b. Find mean of the process x(t). 8c. Find the total power in the process x(t). s 9. Consider a system with transfer function H(s) =. Describe the output, s 1 say y(t), if the input is a random constant, say x(t) = X, where X: N(, = 4). Hint: the output will of course be random. You need to say what type of random process it is, and provide any parameters needed to specify the pdf. 30. Consider the system : x(t) H(f) y(t) where H(f) = 1/(1+j00f) a. Find S Y (f) if x(t) is white noise, with psd S X (f) =. b. Describe y(t) if x(t) = cos(t + ) where is U(0, ). c. Find S X (f) and S Y (f) if R X () = 5 exp(- ). 31. Consider a Gaussian white random process X(t) that is N(0, = 9) for < t < 5, and where. a. Use MATLAB to graph 3 sample functions from this RP, on the range t (0, 6). Use MATLAB s subplot function to plot all 3 sample functions on one page, one underneath another. b. Now assume that the random process is input to a sample-and-hold device, taking samples at times: 0,.,.4,, 5.8. Use MATLAB to plot the output that
ECE 450 7 would result from each of your 3 sample functions generated in part a. Plot the sample-and-hold functions on top of the corresponding sample functions, using MATLAB s subplot function as described in part a. 3. Consider a Gaussian random process X(t) with autocorrelation function shown below: R ( ) 4 X - a. Find the mean square power, E(X ). b. Find the variance of the process. c. Find E(X(3) X(4)) 33. A wide-sense stationary (WSS) random process X(t) has autocorrelation function R x () = 5 exp(-4 ) + 16 cos(0) + 36 Find: a. E[X (t)] b. E[X(t)] c. the variance of the process. 34. x(t) H ( f ) y(t) AWGN, S X (f) = 10 Say H(f) is an ideal LPF with cut-off frequency 1K Hz. (Assume unit-gain over the pass-band.) For the system and input given above, find a. E[X] b. E[X ] c. S y (f) d. E[Y ]