EE 503 : Homework 3 Due : 09/23/2021, Thursday before class

EE 503 : Homework 3 Due : 09/23/2021, Thursday before class. 1. An urn contains n + m balls, of which n are red and m are black. They are drawn from the urn one at a time, without replacement. Let X be the number of red balls removed before the first black ball is chosen. We are interested in determining E[X]. To obtain this quantity, number the red balls from 1 to n. Now define the random variables Xi , i = 1, · · · n, by ? ? 1, if red ball i is taken before any black ball is chosen Xi = ? 0, otherwise a) Express X in terms of the Xi . b) Find E[X]. 2. Let X be a uniformly distributed random variable in the interval [−π, π]. What is the cdf of Y = tan X? 3. If X ∼ N (µ, σ), what is the pdf of Y = (X − µ)2 /σ 2 ? 4. A Cauchy random variable X has the following pdf fX (x) = π(x2 α , −∞ < x < ∞, α > 0 + α2 ) a) Find E[X] and E[X 2 ]? b) Suppose a random variable X had a characteristic function φ(ω) = e−|ω| , what is its mean and variance? 5. Three types of customers arrive at a service station. The times required to service type 1 and type 2 customers are exponential random variables with respective means 1 and 10 seconds. Type 3 customers require a constant service time of 2 seconds. Suppose that the proportion of type 1, 2 and 3 customers is 1/2, 1/8 and 3/8, respectively. Find the probability that an arbitrary customer requires more than 15 seconds of service time. 6. The average score in the final exam of a course is 65 and the standard deviation is 10. a) Give an upper bound on the probability of a student scoring more than 95? b) Suppose the scores follow a normal distribution. Compute the probability of a student scoring more than 95 and compare it to the bound obtained in a). 7. The number X of electrons counted by a receiver in an optical communication system is a Poisson random variable with rate λ1 when a signal is present and with rate λ0 < λ1 when a signal is absent. Suppose that a signal is present with probability p. a) Find P [signal present|X = k] and P [signal absent|X = k]. b) The receiver uses the following decision rule: If P [signal present|X = k] > P [signal absent|X = k], decide signal present; otherwise decide signal absent Show that this decision rule leads to the following threshold rule: If X > T , decide signal present; otherwise, decide signal absent. c) What is the probability of error for the above decision rule? 8. Exponential Random Variable: a) Generate instances of exponential random distribution from a uniform distributed random variable, random.uniform(0,1). b) Use the built-in function random.exponential() to generate the same number of instances. c) Compare the histograms of a and b. 2