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# Author: hwmadeeasy

# 7. (10 point bonus- no partial credit) Based on observations, a ran- dom variable is a believed to have a PDF of the form fX (x) = asin^2 x − pi/2

# 6. (20 points) X (t) is a wide sense stationary process with zero mean and autocorrelation RX (t) = e− a | t | with a = p. We observe Y (t) = X (t) + N (t) where N (t) is a white Gaussian noise process with N0 = 10−5 . X (t) and N (t) are independent. (a) What is the transfer function of the optimum ﬁlter for estimat- ing X (t) given Y (t)? (b) What is the mean square error of the optimum ﬁlter estimate?

# 4. (20 points) A random variable is deﬁned to be Z = X +Y where X and Y are independent and both U ( −1 , 1). (a) Find the PDF of Z. (b) Use the PDF in part (a) to ﬁnd the mean and variance of Z. Note that whilte there are other ways of ﬁnding these values, you will only receive credit for using the PDF of Z.

# 3. (20 points) A binary bit stream with probability of a 1 being p = 0.25 is used to form “supersymbols” of size 3. (a) Find the probability of each 3 bit number and develop a Huff- man code for it. (b) Find the expected value of the length of a word and the en- tropy.

# 2. (20 points) A Poisson random variable X with parameter l = 1 is used to form the random variable Y = aX + b with a = 3 and b = 1. (a) Find the moment generating function for Y. (b) Use the moment generating function from part (a) to ﬁnd the mean of Y, µY and the variance of Y, s2 Y. Note that whilte there are other ways of ﬁnding these values, you will only receive credit for using the moment generating function.

# 1. (20 points) Answer the following concept questions. (a) If two events are mutually exclusive, are they independent as well? (b) If three events form a partition of a sample space, what is true about the probabilities of those three events? (c) In general, is the conditional probability of two events smaller or larger than the joint probability ? (d) I have calculated that the CDF of a random variable X is 0.1 when x = 1 and 0.09 when x = 2. Do you have enough infor- mation to know if I have done this correctly, and if so, did I get it right ? (e) I have two coins. One is fair and one isn’t. Which one will have the higher entropy ?

# 20.) Calculate the current i shown in Figure 12. Place answer in box.

# Questions 18 and 19: Consider the circuit shown in Figure 11. 18.) Find the capacitor voltage, vc ( t), at t = 1 second. Place answer in box. 19.) Find the power being absorbed by the 33 Ω resistor at t = 1 second. Place answer in box.

# Questions 16 and 17: Use the superposition principle to find the components of the current i shown in Figure 10. 16.) Determine the component of i due to the voltage source only and place your answer in the box. 17.) Determine the component of i due to the current source only and place your answer in the box.

# Questions 14 and 15: Consider the circuit shown in Figure 9. Design the circuit to have a time- constant of 3 seconds for t > 0 and vC ( ∞)= 2 V. 14.) Determine the required value of R and place your answer in the box. 15.) Determine the required value of C and place your answer in the box.

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