ECE-340, Spring 2015 Review Questions

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1 ECE-340, Spring 2015 Review Questions 1. Suppose that there are two categories of eggs: large eggs and small eggs, occurring with probabilities 0.7 and 0.3, respectively. For a large egg, the probabilities of having 1, 2, or 3 yolks are 0.95, and 0.005, respectively. On the other hand, for a small egg, the probabilities of having 1, 2, or 3 yolks are 0.98, and 0.001, respectively. Suppose that an egg is picked at random and let X represent the number of yolks in it. a) Plot the cumulative distribution function of X. b) Calculate the mean and variance of X. c) Suppose that it is known that an egg has more than one yolk, what is the probability that is has three yolks? d) Suppose that it is known that an egg has three yolks, what is the probability that it is a large egg? 2. (defer this problem to the end of the last week of class) Consider the RC circuit shown below. Suppose that the noisy resistor is at temperature T Kelvin and its noise is modeled as Johnson noise. + Y(t) - a) Extract a Thevenin equivalent circuit for the this circuit showing the noise source as the input, as well as a noiseless resistor. Precisely write the mean, autocorrelation function and the power spectral density of the random process representing noise source. b) Is the noise process wide-sense stationary? Justify your answer. c) Find the mean and average power of the voltage Y(t) across the capacitor. d) Find the autocorrelation function of the process Y(t). e) Find the noise equivalent bandwidth for this RC circuit 3. Nine resistors are sampled from a large box of resistors. The actual resistance values of these nine elements are: 3.9, 3.7, 3.7, 3.8, 3.5, 3.9, 4.0, 3.7, 4.2, all in the units of kω. a) Find the sample mean. You must show all your work. b) Write out the formula for calculating the sample variance but do not evaluate it. c) Suppose that the true mean and true variance of the resistors are 4 kω and 0.2 kω 2, respectively. Suppose now that 100 resistors are drawn at random from the box, and let denote the sample mean corresponding to the 100 samples. Use Chebychev's inequality to estimate the probability that the sample mean (corresponding to the 100 samples) is NOT within 10% of the true mean. You must show all your work. d) How large must the sample size be in order for the sample mean to be within 10% of the true mean with probability 0.99? You must show all your work. e) Suppose that in actuality the value of a randomly selected resistor from the large box is modeled as a Gaussian random variable, X, with true means and variances as given above. Derive an expression for the probability that the sample mean (corresponding to the 100 selected samples) is not within 10% of the true mean. You can leave your answer in terms of either the Φ function or the erf function. 1

2 f) Suppose that we are required to find the confidence interval corresponding to a confidence level of q=0.99 for a sample size of 100. Namely, we want the interval I=[-a +μ, a + μ] such that P{ I}= q. Find an expression for the unknown a. Leave your answer in terms of either the Φ 1 function or the erf 1 function. g) Use the central limit theorem to approximately calculate the probability in Part (c), then compare your results to the Chebychev bound obtained in Parts (c) and (e). Comment on your results. h) Repeat Part (f) for q=0.9 and Comment on your results. i) Repeat Part (f) while changing the sample size to 50 and 200. Comment on your results. 4. Suppose that X is a random variable that is uniformly distributed in [0, 1]. Define the new random variables Y = 3 X -1.5, Z = X 5 and W = u(x - 0.2), where u is the unit-step function. a) Determine and plot the pdf of Y. b) Determine the pdf of Z. c) Determine and plot the cumulative distribution function of W. 5. A random variable Y has the cumulative distribution function shown below. It is known that P{Y=2}= 0.6. Plot the pdf of Y and calculate the F Y (y) variance of Y y 6. Let N 1 represent the number of hits that a website receives in the interval [0,1], and let N 2 represent the number of hits that a website receives in the interval (1,3]. Assume that N 1 and N 2 are Poisson random variables mean values 10 and 20, respectively. Let N= N 1 + N 2 represent the total number of hits in the interval [0,3]. a) Find the correlation coefficient between N 1 and N. b) Find the variance of N. c) Calculate E[N N 1 =9]. 7. A pair of random variables, X and Y, have a joint pdf given by f XY (x,y)=x+y, for 0 x 1, 0 y 1, and f XY (x,y)=0 otherwise. Calculate P{0 < X < 0.1 Y=0.5}. Show all your work. 8. Let X be a uniform random variable in the interval [0,1], and let Y be an exponentially distributed random variable with a mean value of 2. Assume that X and Y are independent. Find and plot the probability density function of Z=X+Y. 9. A football stadium has 1000 light bulbs used to illuminate the field. The probability that each light bulb is faulty when the switch is closed is Use the central-limit theorem to 2

3 estimate the probability that the total number of working light bulbs (after the switch is closed) is greater than 900? Be precise and state any assumptions you make. 10 (a). State the central limit theorem. Be precise. (b) In a certain town, the average age of an individual is known to be 40 years and the standard deviation is 6 years. Suppose that 50 individuals are selected at random (independently) from the population to participate in a survey. You are asked to use the central limit theorem to approximately calculate the probability that the sum of the ages of the 50 individuals exceeds Express your answer using the Q function. (c) Suppose that your answer to part b is p, what is the probability that the sample mean of the 50 selected individuals exceeds 50? [Hint: Before you do anything relate this question to part b.] 11. (a) State Chebychev s inequality. (b) Refer to Problem 10. Use Chebychev s inequality to estimate the probability that the sample mean of the 50 individuals is within ±5 years from the true mean of 40 years. 12. Consider two urns; the first urn contains n+1 balls, numbered from 0 to n. The second urn contains n+3 balls numbered 0 to n+2. A ball is drawn at random from the first urn, another ball is drawn from the second urn, and their values are added. Denote this sum by the random variable N. Assume that the numbers obtained from two urns are independent. Find and plot the probability mass function of N. State any assumptions you ve made. 13. Suppose that X is a random variable that is Gaussian with a mean of -1 and a variance 2. Define the new random variables Y = 0.5 X +1 and Z = g(x), where 1 x 1 gx ( ) = a. Determine the pdf of Y. 0 x < 1 b. Determine and plot the probability mass function of Z. 14. (a) Derive the characteristic function of a Poisson random variable with mean 3. 3

4 (b) Use your answer in part a to prove that the sum of two independent Poisson random variables is also a Poisson random variable. (c) Use your answer in part (a) to calculate E[N 2 ]. (d) Describe a physical example for which a Poisson random variable would be a good model for and interpret the significance of the conclusion in part b. 15. Suppose that the mean and variance of the random variable X are 3 and 8, respectively; what is the variance of the random variable 3X + 5? 16. Suppose that X and Y are independent Gaussian rvs with means 1 and -5, respectively, and variances 2 and 7, respectively. Let Z be defined as Z=2X+3Y. Find the pdf of Z. Hint: do not attempt to use convolutions. 17. Suppose that X and Y are independent Poisson rvs with means 1.5 and 2.6, respectively Let Z be defined as Z=X+Y. Find the pdf of Z. Hint: do not attempt to use convolutions. 4

6 Problem 2 Consider the RC circuit shown below. Suppose that the resistor is at temperature T Kelvin exhibiting Johnson noise. Solution: a) + X(t) + + Y(t) The input is modeled as white noise with autocorrelation function R XX (t 1,t 2 )= 2Rk B T ( t 2 - t 1 ) and power spectral density S XX ( )=2Rk B T. Also, we know that E[X(t)]=0. b) Yes the noise is WSS since the mean is time independent and the autocorrelation function is dependent only on time difference. c) We first find the transfer function. By using phasor analysis, we obtain H(j )= Y(j )/ X(j ) = (1/j C)/(R + 1/j C) = 1 / (1+ j RC) = (1/RC) / [(1/RC)+ j ]. Note that the impulse response of this circuit is h(t) = F 1 {H(j )} = (1/RC) exp( t/rc) u(t). Now E[Y(t)]= E[X(t)] = 0, since E[X(t)]=0. Next, E[Y 2 (t)]= R XX ( )= (1/2 ), where S YY ( )= H(j ) 2 S XX ( ) = (1/RC) 2 / [(1/RC) 2 + ] 2Rk B T. To perform the integration, we recall the following integral: which we can use to obtain /2, E[Y 2 (t)]= (1/2 ) = 2 = /. (W) d) To find the autocorrelation function of Y(t), we find the inverse Fourier transform of S YY ( ). Recall that in class we derived the following Fourier transform pair: F 1 { } = e -a t, where a >0. With this knowledge, we find R YY ( ) = (k B T/C) e - RC Note that R YY ( ) = (k B T/C), which is in agreement with the answer in part (c). e) [Optional] The equivalent bandwidth is the bandwidth (in Hertz), B N, of an ideal low pass filter (with constant height equal to unity) that admits the same average power as the filter H. Thus, we set 2Rk B T 2 B N = (k B T/C), which yields B N = 1/4RC. 2

for valid PSD. PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT I

for valid PSD. PART B (Answer all five units, 5 X 10 = 50 Marks) UNIT I Code: 15A04304 R15 B.Tech II Year I Semester (R15) Regular Examinations November/December 016 PROBABILITY THEY & STOCHASTIC PROCESSES (Electronics and Communication Engineering) Time: 3 hours Max. Marks: