Show that for a Gaussian distribution and a zero mean (2* = 0) from the reference...
Show that for a Gaussian distribution and a zero mean (2* = 0) from the reference line (determined by the M system) the theoretical skewness is zero and the kurtosis is 3.
Homework Answers
Skewness is zero and kurtosis =3 means that the distribution is follow a normal distribution with mean mu and variance sigma^2. According to your question, the mean is deviated from the reference line with value 0. Hence, the distribution is follow a Gaussian distribution with mean zero and variance signa^2. Here, the plot of Gaussian distribution with mean zero and sigma^2=1.
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Show that for a Gaussian distribution and a zero mean (2* = 0)
from the reference...
4.48 A Gaussian random variable has mean μ and variance σ2 (a) Show that the moment geneng fnctio...
Please solve this. Thank you.
4.48 A Gaussian random variable has mean μ and variance σ2 (a) Show that the moment geneng fnction (MGF) for the Gaussian ran dom variable is given by Hint: Use the technique of "completing the square. b) Assume that 0 and use the MGF to compute the first four moments of x a well hvarian, sks, and kurtosis. (c) What are the mean, variance, skewness, and kurtosis for μ 0?
4.48 A Gaussian random variable...
Let X and Y be two independent Gaussian random variables with common variance σ2. The mean of X i...
Let X and Y be two independent Gaussian random variables with common variance σ2. The mean of X is m and Y is a zero-mean random variable. We define random variable V as V- VX2 +Y2. Show that: 0 <0 Where er cos "du is called the modified Bessel function of the first kind and zero order. The distribution of V is known as the Ricean distribution. Show that, in the special case of m 0, the Ricean distribution simplifies...
2. Generate a large number of Gaussian distributed random numbers with mean 0 and variance 1....
2. Generate a large number of Gaussian distributed random numbers with mean 0 and variance 1. (HINT: use the randn command) a) Provide a scatter plot of the random numbers b) Plot the pdf of the distribution (similar to 1) and compare with theoretical pdf given in the class handout. c) If I want to generate a Gaussian distributed random numbers with mean 2 and variance 9 from the previously generated set of random numbers, how would I do it?...
2. Suppose that we are attempting to estimate the mean of a Gaussian distribution describing a se...
2. Suppose that we are attempting to estimate the mean of a Gaussian distribution describing a set of scalar measurements, x. Assume that the variance of the distribution is known to be,02-9. (a) Suppose that we have some independent sample measurements of the sensor output as shown below. What is the maximum likelihood estimate of the mean? Samples 7, 4, 7, 3, 1,-1, 3, 5, 2, 1 (b) Suppose based on some prior measurements, we have a prior distribution on...
A communication system has a Gaussian pdf with zero mean, o = 1/2 variance and a...
A communication system has a Gaussian pdf with zero mean, o = 1/2 variance and a bandwidth W = 10kHz with noise power No 10-15 W/Hz. The transmission loss power is 100dB. Calculate the transmitted power, St, to achieve the output signal to noise ratio, (Sp)... = 40 dB for each modulation scheme as listed below. Show your all calculations for each modulation scheme in detail and explain your solutions clearly. SSB b. DSB-LC with modulation index u = 1...
Problem 5 of 5Sum of random variables Let Mr(μ, σ2) denote the Gaussian (or normal) pdf with Inea...
Problem 5 of 5Sum of random variables Let Mr(μ, σ2) denote the Gaussian (or normal) pdf with Inean ,, and variance σ2, namely, fx (x) = exp ( 2-2 . Let X and Y be two i.i.d. random variables distributed as Gaussian with mean 0 and variance 1. Show that Z-XY is again a Gaussian random variable but with mean 0 and variance 2. Show your full proof with integrals. 2. From above, can you derive what will be the...
Suppose that X is a Gaussian Random Variable with zero mean and unit variance. Let Y=aX3...
Suppose that X is a Gaussian Random Variable with zero mean and unit variance. Let Y=aX3 + b, a > 0 Determine and plot the PDF of Y
The input to a system is a Gaussian random variable below X with zero mean and variance of σ- as ...
The input to a system is a Gaussian random variable below X with zero mean and variance of σ- as shown x System The output of the system is a random variable Y given as follows: -a b, X>a (a) Determine the probability density function of the output Y (b) Now assume that the following random variable is an input to the system at time t: where the amplitude A is a constant and phase s uniformly distributed over (0,2T)....
7. Let Z be Gaussian white noise, i.e. Z is a sequence of i.i.d. normal r.v.s each with mean zero...
7. Let Z be Gaussian white noise, i.e. Z is a sequence of i.i.d. normal r.v.s each with mean zero and variance 1. Define Zt, t(-1- 1)/v2, if t is odd Show that Xis WN(0,1) (that is, variables Xt and Xt+k,k2 1, are uncorrelated with mean zero and variance 1) but that Xt and Xi-i are not i.i.d
7. Let Z be Gaussian white noise, i.e. Z is a sequence of i.i.d. normal r.v.s each with mean zero and variance...
Let X be Gaussian with zero mean and unit variance. Let Y = |X|. Find: a)...
Let X be Gaussian with zero mean and unit variance. Let Y = |X|. Find: a) The PDF fY (y) b) The mean E[Y ] c) Here X is uniform in (0, 1), but now you are asked to find a functiong(·) such that the PDF of Y = g(X) is ?2y 0≤y<1fY (y) = 0 otherwise
Please solve this. Thank you.
4.48 A Gaussian random variable has mean μ and variance σ2 (a) Show that the moment geneng fnction (MGF) for the Gaussian ran dom variable is given by Hint: Use the technique of "completing the square. b) Assume that 0 and use the MGF to compute the first four moments of x a well hvarian, sks, and kurtosis. (c) What are the mean, variance, skewness, and kurtosis for μ 0?
4.48 A Gaussian random variable...
Let X and Y be two independent Gaussian random variables with common variance σ2. The mean of X is m and Y is a zero-mean random variable. We define random variable V as V- VX2 +Y2. Show that: 0 <0 Where er cos "du is called the modified Bessel function of the first kind and zero order. The distribution of V is known as the Ricean distribution. Show that, in the special case of m 0, the Ricean distribution simplifies...
2. Generate a large number of Gaussian distributed random numbers with mean 0 and variance 1. (HINT: use the randn command) a) Provide a scatter plot of the random numbers b) Plot the pdf of the distribution (similar to 1) and compare with theoretical pdf given in the class handout. c) If I want to generate a Gaussian distributed random numbers with mean 2 and variance 9 from the previously generated set of random numbers, how would I do it?...
2. Suppose that we are attempting to estimate the mean of a Gaussian distribution describing a set of scalar measurements, x. Assume that the variance of the distribution is known to be,02-9. (a) Suppose that we have some independent sample measurements of the sensor output as shown below. What is the maximum likelihood estimate of the mean? Samples 7, 4, 7, 3, 1,-1, 3, 5, 2, 1 (b) Suppose based on some prior measurements, we have a prior distribution on...
A communication system has a Gaussian pdf with zero mean, o = 1/2 variance and a bandwidth W = 10kHz with noise power No 10-15 W/Hz. The transmission loss power is 100dB. Calculate the transmitted power, St, to achieve the output signal to noise ratio, (Sp)... = 40 dB for each modulation scheme as listed below. Show your all calculations for each modulation scheme in detail and explain your solutions clearly. SSB b. DSB-LC with modulation index u = 1...
Problem 5 of 5Sum of random variables Let Mr(μ, σ2) denote the Gaussian (or normal) pdf with Inean ,, and variance σ2, namely, fx (x) = exp ( 2-2 . Let X and Y be two i.i.d. random variables distributed as Gaussian with mean 0 and variance 1. Show that Z-XY is again a Gaussian random variable but with mean 0 and variance 2. Show your full proof with integrals. 2. From above, can you derive what will be the...
The input to a system is a Gaussian random variable below X with zero mean and variance of σ- as shown x System The output of the system is a random variable Y given as follows: -a b, X>a (a) Determine the probability density function of the output Y (b) Now assume that the following random variable is an input to the system at time t: where the amplitude A is a constant and phase s uniformly distributed over (0,2T)....
7. Let Z be Gaussian white noise, i.e. Z is a sequence of i.i.d. normal r.v.s each with mean zero and variance 1. Define Zt, t(-1- 1)/v2, if t is odd Show that Xis WN(0,1) (that is, variables Xt and Xt+k,k2 1, are uncorrelated with mean zero and variance 1) but that Xt and Xi-i are not i.i.d
7. Let Z be Gaussian white noise, i.e. Z is a sequence of i.i.d. normal r.v.s each with mean zero and variance...
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