Question

Consider the given sequence defined as x[n] = A \cdot n + w[n], where w[n] denotes independent and identically distributed (idd) random variables following an Erlang distribution. The probability density function (PDF) of w[n] is given by p(w[n]; \lambda) = \lambda^2 w[n]e^{-\lambda w[n]} for w[n] \ge 0 and \lambda \ge 0. (Q1) Theoretical Plotting: Use MATLAB to theoretically plot the PDF of x by setting \lambda to 3 and A to 10. (Q2) Data Generation: Employ MATLAB to generate a dataset of 10000 or more samples of x[n] and plot the normalized histogram. Ensure that the generated histogram aligns with the theoretical PDF obtained in Q1. (Q3) Log-Likelihood Analysis: Plot the Log-Likelihood function for the parameter \lambda based on the generated dataset. Observe the presence of a peak at \lambda = 3. (Q4) Variance Analysis: Repeat the previous step for the variance of x[n], which is given by 2\lambda^2. Confirm the existence of a peak at 2/9.

          Consider the given sequence defined as x[n] = A \cdot n + w[n], where w[n] denotes independent and identically distributed (idd) random variables following an Erlang distribution. The probability density function (PDF) of w[n] is given by p(w[n]; \lambda) = \lambda^2 w[n]e^{-\lambda w[n]} for w[n] \ge 0 and \lambda \ge 0.
(Q1) Theoretical Plotting: Use MATLAB to theoretically plot the PDF of x by setting \lambda to 3 and A to 10.
(Q2) Data Generation: Employ MATLAB to generate a dataset of 10000 or more samples of x[n] and plot the normalized histogram. Ensure that the generated histogram aligns with the theoretical PDF obtained in Q1.
(Q3) Log-Likelihood Analysis: Plot the Log-Likelihood function for the parameter \lambda based on the generated dataset. Observe the presence of a peak at \lambda = 3.
(Q4) Variance Analysis: Repeat the previous step for the variance of x[n], which is given by 2\lambda^2. Confirm the existence of a peak at 2/9.

Consider the given sequence defined as x[n] = A ·n + w[n], where w[n] denotes independent and identically distributed (idd) random variables following an Erlang distribution. The probability density function (PDF) of w[n] is given by p(w[n]; λ) = λ^2 w[n]e^-λw[n] for w[n] ≥0 and λ≥0.
(Q1) Theoretical Plotting: Use MATLAB to theoretically plot the PDF of x by setting λto 3 and A to 10.
(Q2) Data Generation: Employ MATLAB to generate a dataset of 10000 or more samples of x[n] and plot the normalized histogram. Ensure that the generated histogram aligns with the theoretical PDF obtained in Q1.
(Q3) Log-Likelihood Analysis: Plot the Log-Likelihood function for the parameter λbased on the generated dataset. Observe the presence of a peak at λ= 3.
(Q4) Variance Analysis: Repeat the previous step for the variance of x[n], which is given by 2λ^2. Confirm the existence of a peak at 2/9.

Added by Madeline T.

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