Question 1)
Consider the following Cascade of two Binary symmetric channels (CBSC) with probabilities as indicated in the figure below
1. Find P(Y=1 / X=1 ), P(Y=0 / X=1)
2. Find P(Y=1 / X=0 ), P(Y=0 / X=0)
3. Find The Channel Matrix for
each BSC separately
4. Find The overall
Channel Matrix of the cascade channels
5. Assume that P1 = P2 =
Pe , Prove that the Channel Matrix is
M2
6. Use the assumptions and results in part (5) to
find the channel matrix for M3
7. Find the overall
Pefrom the channel matrix in case of
M3
------------------------------------------------------------------------------------------------------------------------
Question 2)
Consider the following probability density function PX(X) described
as shown in the figure below
1. Find the mean of the random
variable x
2. Find the mean square of the
random variable x
3. Find the variance of the
random variable x
IK 3 X-
Homework Answers
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Question 1) Consider the following Cascade of two Binary symmetric channels (CBSC) with probabilities as indicated in the figure below 1. Find P(Y=1 / X=1 ), P(Y=0 / X=1) 2. Find P(Y=1 / X=0 ), P(Y=0...
Let us consider a binary symmetric channel, as shown in Figure 1, where the probabilities of the input X are Pr(X-0] =...
Let us consider a binary symmetric channel, as shown in Figure 1, where the probabilities of the input X are Pr(X-0] = m and Pr(X-1-1-m, and the error probability during the transmission from X and Y is p. 0 1-p Figure 1: A typical binary symmetric channel, where the input is X and the output is Y. a) Given that p-1/3 and m-3/4, find H(X), H (Y), H (YİX), and 1(X:Y). (8 marks) b) Still given p = 1 /3....
Q2) Consider the following probability density function PX(X) described as shown in the figure below 1....
Q2)
Consider the following probability density function PX(X) described
as shown in the figure below
1. Find the mean of the random
variable x
2. Find the mean square of the
random variable x
3. Find the variance of the
random variable x
IK 3 X-
(1 Consider the symmetric matrix A = 2 10 2 0 2 2 1. Answer the...
(1 Consider the symmetric matrix A = 2 10 2 0 2 2 1. Answer the following questions. 2 3 (1) Find the eigenvalues , , and iz (2 <, <1z) of the matrix A and their corresponding eigenvectors. (2) Find the orthogonal matrix B and its inverse matrix B' that satisfy the following equation: (4 0 0 B-'AB = 0 0 lo o 2) (3) Suppose that the real vectors y and 9 satisfy the following relationship: Show that...
Consider the random variables X and Y with joint density function [5] f(x,y)=1/x , 0<y<x<1 i) Find P(X > 0 ....
Consider the random variables X and Y with joint density function [5] f(x,y)=1/x , 0<y<x<1 i) Find P(X > 0 . 5 , y >0.5). ii) Find fX | y(x) and fY | x(y)..
2) Consider a random variable with the following probability distribution: P(X-0)-0., Px-1)-0.2, PX-2)-0.3, PX-3) -0.3, and PX-4)-0.1 A. Generate 400 values of this random variable with the given pro...
2) Consider a random variable with the following probability distribution: P(X-0)-0., Px-1)-0.2, PX-2)-0.3, PX-3) -0.3, and PX-4)-0.1 A. Generate 400 values of this random variable with the given probability distribution using simulation. B. Compare the distribution of simulated values to the given probability distribution. Is the simulated distribution indicative of the given probability distribution? Explain why or why not. C. Compute the mean and standard deviation of the distribution of simulated values. How do these summary measures compare to the...
Consider the following joint probability distribution on the random variables X and Y given in matrix...
Consider the following joint probability distribution on the random variables X and Y given in matrix form by Pxy P11 P12 P13 PXY-IP21 p22 p23 P31 P32 P33 P41 P42 P43 HereP(i, j) P(X = z n Y-J)-Pu represents the probability that X-1 and Y = j So for example, in the previous problem, X and Y represented the random variables for the color ([Black, Red]) and utensil type (Pencil,Pe pblackpen P(X = Black Y = Pen) = P(Black n...
The Bernoulli random variable takes values 0 and 1, and has probability function f(x) = (p^x) [(1 − p)^(1−x)]
The Bernoulli random variable takes values 0 and 1, and has probability function f(x) = px (1 − p)1−x(a) By calculating f(0) and f(1), give a practical example of a Bernoulli experiment, and a Bernoulli random variable. (b) Calculate the mean and variance of the Bernoulli random variable.
decimal. I variable. Find P(X<0). Express your answer as a 0.5 Question 2 1 pts Let...
decimal. I variable. Find P(X<0). Express your answer as a 0.5 Question 2 1 pts Let X be a standard normal random variable. Find PIX <-2.27). Express your answer as a decimal. Question 3 1 pts Let X be a standard normal random variable. Find PX 2.82), Express your answer as a
1. Consider two random variables X and Y with joint density function f(x, y)-(12xy(1-y) 0<x<1,0<p<1 otherwise...
1. Consider two random variables X and Y with joint density function f(x, y)-(12xy(1-y) 0<x<1,0<p<1 otherwise 0 Find the probability density function for UXY2. (Choose a suitable dummy transformation V) 2. Suppose X and Y are two continuous random variables with joint density 0<x<I, 0 < y < 1 otherwise (a) Find the joint density of U X2 and V XY. Be sure to determine and sketch the support of (U.V). (b) Find the marginal density of U. (c) Find...
(1) Suppose the pdf of a random variable X is 0, otherwise. (a) Find P(2 <...
(1) Suppose the pdf of a random variable X is 0, otherwise. (a) Find P(2 < X < 3). (b) Find P(X < 1). (e) Find t such that P(X <t) = (d) After the value of X has been observed, let y be the integer closest to X. Find the PMF of the random variable y U (2) Suppose for constants n E R and c > 0, we have the function cr" ifa > 1 0, otherwise (a)...
Let us consider a binary symmetric channel, as shown in Figure 1, where the probabilities of the input X are Pr(X-0] = m and Pr(X-1-1-m, and the error probability during the transmission from X and Y is p. 0 1-p Figure 1: A typical binary symmetric channel, where the input is X and the output is Y. a) Given that p-1/3 and m-3/4, find H(X), H (Y), H (YİX), and 1(X:Y). (8 marks) b) Still given p = 1 /3....
Q2)
Consider the following probability density function PX(X) described
as shown in the figure below
1. Find the mean of the random
variable x
2. Find the mean square of the
random variable x
3. Find the variance of the
random variable x
IK 3 X-
(1 Consider the symmetric matrix A = 2 10 2 0 2 2 1. Answer the following questions. 2 3 (1) Find the eigenvalues , , and iz (2 <, <1z) of the matrix A and their corresponding eigenvectors. (2) Find the orthogonal matrix B and its inverse matrix B' that satisfy the following equation: (4 0 0 B-'AB = 0 0 lo o 2) (3) Suppose that the real vectors y and 9 satisfy the following relationship: Show that...
2) Consider a random variable with the following probability distribution: P(X-0)-0., Px-1)-0.2, PX-2)-0.3, PX-3) -0.3, and PX-4)-0.1 A. Generate 400 values of this random variable with the given probability distribution using simulation. B. Compare the distribution of simulated values to the given probability distribution. Is the simulated distribution indicative of the given probability distribution? Explain why or why not. C. Compute the mean and standard deviation of the distribution of simulated values. How do these summary measures compare to the...
Consider the following joint probability distribution on the random variables X and Y given in matrix form by Pxy P11 P12 P13 PXY-IP21 p22 p23 P31 P32 P33 P41 P42 P43 HereP(i, j) P(X = z n Y-J)-Pu represents the probability that X-1 and Y = j So for example, in the previous problem, X and Y represented the random variables for the color ([Black, Red]) and utensil type (Pencil,Pe pblackpen P(X = Black Y = Pen) = P(Black n...
decimal. I variable. Find P(X<0). Express your answer as a 0.5 Question 2 1 pts Let X be a standard normal random variable. Find PIX <-2.27). Express your answer as a decimal. Question 3 1 pts Let X be a standard normal random variable. Find PX 2.82), Express your answer as a
1. Consider two random variables X and Y with joint density function f(x, y)-(12xy(1-y) 0<x<1,0<p<1 otherwise 0 Find the probability density function for UXY2. (Choose a suitable dummy transformation V) 2. Suppose X and Y are two continuous random variables with joint density 0<x<I, 0 < y < 1 otherwise (a) Find the joint density of U X2 and V XY. Be sure to determine and sketch the support of (U.V). (b) Find the marginal density of U. (c) Find...
(1) Suppose the pdf of a random variable X is 0, otherwise. (a) Find P(2 < X < 3). (b) Find P(X < 1). (e) Find t such that P(X <t) = (d) After the value of X has been observed, let y be the integer closest to X. Find the PMF of the random variable y U (2) Suppose for constants n E R and c > 0, we have the function cr" ifa > 1 0, otherwise (a)...
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