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Problem 1 A Poisson process is a continuous-time discrete-valued random process, X(t), that counts the number of events...
Homogeneous Poisson process N(t) counts events occurring in a time interval and is characterized by Ņ(0)-0...
Homogeneous Poisson process N(t) counts events occurring in a time interval and is characterized by Ņ(0)-0 and (t + τ)-N(k) ~ Poisson(λτ), where τ is the length of the interval (a) Show that the interarrival times to next event are independent and exponentially distributed random variables (b) A random variable X is said to be memoryless if P(X 〉 s+ t | X 〉 t) = P(X 〉 s) y s,t〉0. that this property applies for the interarrival times if...
Exercises 3-8 all refer to events occurring in time according to a Poisson process with parameter...
Exercises 3-8 all refer to events occurring in time according to a Poisson process with parameter λ on 0 š t < oo. Here x(t) denotes the number of events that occur in the time interval (0, t] 3 Find the conditional probability that there are m events in the first s units of time, given that there are n events in the first t units of time, where 0 s m < n and 0 s s < t.
P9.3 A random process X(t) has the following member functions: x1 (t) -2 cos(t), x2(t)2 sin(t),...
P9.3 A random process X(t) has the following member functions: x1 (t) -2 cos(t), x2(t)2 sin(t), x3(t)- 2 (cos(t) +sin(t)),x4t)cost) - sin(t), xst)sin(t) - cos(t).Each member function occurs with equal probability. (a) Find the mean function, Hx (t). (b) Find the autocorrelation function, Rx(t1,t2) (c) Is this process WSS? Is it stationary in the strict sense?
Let (N(t) 0 be a Poisson process with rate A> 0, and suppose to 0 < t, < t2 < ... are the successive occurr...
Let (N(t) 0 be a Poisson process with rate A> 0, and suppose to 0 < t, < t2 < ... are the successive occurrence times of events in the process. Prove that the interarrival times Sn t-t are independent and identically distributed according to Exponential(A)
Let (N(t) 0 be a Poisson process with rate A> 0, and suppose to 0
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability...
Recall that a discrete random variable X has Poisson
distribution with parameter λ if the probability mass function of
X
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
2. Let Y(t) = (x(0)+)(\pi) where X(t) is a Poisson process with autocorrelation function Rxx(t1, tz)...
2. Let Y(t) = (x(0)+)(\pi) where X(t) is a Poisson process with autocorrelation function Rxx(t1, tz) = tīta + min(tı, tz), and 6 ~ U(0,2%) is independent of X(t). a. Is X(t) W.S.S.? b. If so, find its power spectral density. [25]
2. Consider the random process x(t) defined by x(t) a cos(wt 6), where w and 0 are constants, and a is a random variable uniformly distributed in the range (-A, A). a. Sketch the ensemble (sample...
2. Consider the random process x(t) defined by x(t) a cos(wt 6), where w and 0 are constants, and a is a random variable uniformly distributed in the range (-A, A). a. Sketch the ensemble (sample functions) representing x(t). (2.5 points). b. Find the mean and variance of the random variable a. (5 points). c. Find the mean of x(t), m(t) E((t)). (5 points). d. Find the autocorrelation of x(t), Ra (t1, t2) E(x (t)x2 )). (5 points). Is the...
2. Let Y(t) = ei(x(0)+o)(\pi) where X(t) is a Poisson process with autocorrelation function Rxx(t1, tz)...
2. Let Y(t) = ei(x(0)+o)(\pi) where X(t) is a Poisson process with autocorrelation function Rxx(t1, tz) = tīta + min(tı, tz), and 6 ~ U(0,2") is independent of X(t). a. Is Y(t) W.S.S.? b. If so, find its power spectral density. [25]
and is X(t) a WSS process? 6.11 Sinusoid with random phase. Consider a random process x(t)-A...
and is X(t) a WSS process?
6.11 Sinusoid with random phase. Consider a random process x(t)-A cos(wot + ?), where wo are nonrandom positive constants and o is a RV uniformly distributed over A and (0, ?), i.e., ? ~11(0, ?). (a) Find the mean function 2(t) of X(t).
Let N(t), t 2 0} be a Poisson process with rate X. Suppose that, for a...
Let N(t), t 2 0} be a Poisson process with rate X. Suppose that, for a fixed t > 0, N (t) Please show that, for 0 < u < t, the number of events that have occurred at or prior to u is binomial with parameters (n, u/t). That is, n. That is, we are given that n events have occurred by time t C) EY'C)" n-i u P(N(u) iN (t)= n) - for 0in
Let N(t), t 2...
Homogeneous Poisson process N(t) counts events occurring in a time interval and is characterized by Ņ(0)-0 and (t + τ)-N(k) ~ Poisson(λτ), where τ is the length of the interval (a) Show that the interarrival times to next event are independent and exponentially distributed random variables (b) A random variable X is said to be memoryless if P(X 〉 s+ t | X 〉 t) = P(X 〉 s) y s,t〉0. that this property applies for the interarrival times if...
Exercises 3-8 all refer to events occurring in time according to a Poisson process with parameter λ on 0 š t < oo. Here x(t) denotes the number of events that occur in the time interval (0, t] 3 Find the conditional probability that there are m events in the first s units of time, given that there are n events in the first t units of time, where 0 s m < n and 0 s s < t.
P9.3 A random process X(t) has the following member functions: x1 (t) -2 cos(t), x2(t)2 sin(t), x3(t)- 2 (cos(t) +sin(t)),x4t)cost) - sin(t), xst)sin(t) - cos(t).Each member function occurs with equal probability. (a) Find the mean function, Hx (t). (b) Find the autocorrelation function, Rx(t1,t2) (c) Is this process WSS? Is it stationary in the strict sense?
Let (N(t) 0 be a Poisson process with rate A> 0, and suppose to 0 < t, < t2 < ... are the successive occurrence times of events in the process. Prove that the interarrival times Sn t-t are independent and identically distributed according to Exponential(A)
Let (N(t) 0 be a Poisson process with rate A> 0, and suppose to 0
Recall that a discrete random variable X has Poisson
distribution with parameter λ if the probability mass function of
X
Recall that a discrete random variable X has Poisson distribution with parameter λ if the probability mass function of X is r E 0,1,2,...) This distribution is often used to model the number of events which will occur in a given time span, given that λ such events occur on average a) Prove by direct computation that the mean of...
2. Let Y(t) = (x(0)+)(\pi) where X(t) is a Poisson process with autocorrelation function Rxx(t1, tz) = tīta + min(tı, tz), and 6 ~ U(0,2%) is independent of X(t). a. Is X(t) W.S.S.? b. If so, find its power spectral density. [25]
2. Consider the random process x(t) defined by x(t) a cos(wt 6), where w and 0 are constants, and a is a random variable uniformly distributed in the range (-A, A). a. Sketch the ensemble (sample functions) representing x(t). (2.5 points). b. Find the mean and variance of the random variable a. (5 points). c. Find the mean of x(t), m(t) E((t)). (5 points). d. Find the autocorrelation of x(t), Ra (t1, t2) E(x (t)x2 )). (5 points). Is the...
2. Let Y(t) = ei(x(0)+o)(\pi) where X(t) is a Poisson process with autocorrelation function Rxx(t1, tz) = tīta + min(tı, tz), and 6 ~ U(0,2") is independent of X(t). a. Is Y(t) W.S.S.? b. If so, find its power spectral density. [25]
and is X(t) a WSS process?
6.11 Sinusoid with random phase. Consider a random process x(t)-A cos(wot + ?), where wo are nonrandom positive constants and o is a RV uniformly distributed over A and (0, ?), i.e., ? ~11(0, ?). (a) Find the mean function 2(t) of X(t).
Let N(t), t 2 0} be a Poisson process with rate X. Suppose that, for a fixed t > 0, N (t) Please show that, for 0 < u < t, the number of events that have occurred at or prior to u is binomial with parameters (n, u/t). That is, n. That is, we are given that n events have occurred by time t C) EY'C)" n-i u P(N(u) iN (t)= n) - for 0in
Let N(t), t 2...
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