# Questions tagged [random]

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74 views

### If $X$ and $Y$ are zero mean independent Gaussian random variables with different variances, what is the density of $\sqrt{X^2+Y^2}$

Let $X\sim\mathcal{N}(0,\sigma_X^2)$ and $Y\sim\mathcal{N}(0,\sigma_Y^2)$ be independent Gaussian random variables. What will be PDF of $Z=\sqrt{X^2+Y^2}$ and $W=\arctan{\left(\frac{Y}{X}\right)}$. ...
23 views

### Proving that this process is weakly-stationary [duplicate]

Let $X(t) = Acos(2\pi f_c t)$ be a random process where $A$ is a uniform random variable within $(-1,1)$. I'm trying to prove this is a weakly(i.e. wide sense) stationary process. I need to show two ...
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### Why the requirement of the GCD of the lengths of all circuits in the graph being one?

I am reading A Mathematical Theory of Communication. The second requirement of an ergodic process confuses me (emphasis mine): All the examples of artificial languages given above are ergodic. This ...
60 views

### Probability of error for detection problem

Let $X \in \mathbb{R}^N$ and $Z \sim \mathcal{N}(0, \sigma^2 I)$ be random vectors. $Y = X + Z$ $X$ can be either $a_0 \in \mathbb{R}^N$ or $a_1\in \mathbb{R}^N$ with equal probability. So the ...
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### Variance of function of random variable

Is their an easier way to find variance of function of random variable? Till now what I am doing is first find probability density function of (function of random variable) then integrate over range.
50 views

### Independence of Functions of random Variable

Consider I am given two functions of one random variable each for example x=cos(at),y=rect(bt) where a and b are random variables.And I am given Probability density function for a and b then if I am ...
73 views

### Random Process at a particular time instance

I was studying Random Process and I thought I understood what it was all about until I came across this example. Consider a random experiment of tossing a coin with sample space S = {H, T} The sample ...
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### How to randomly generate a realistic hourly temperature series given historic data

I have 20 years of historic hourly temperature data in Omaha. I want, given just a seed, to be able to generate a pretty realistic year of hourly temperatures. (I have a program that takes such ...
72 views

### Doubts and some confusion on variance for complex rv

This question is in continuation of the one asked here. Let's say that the measurement noise $w$ or any random variable is circularly Gaussian complex. If the imaginary and real components each has ...
251 views

### How can I calculate the entropy of a signal that's not independent from itself?

I really don't know how to word this better, sorry if that title is confusing. Basically, I'm wanting to use the noise from an AM radio audio signal to generate random numbers. To do this effectively,...
795 views

I have a question about simulating Rayleigh Fading channel. I know theory behind the Rayleigh channel and it is sum of two Gaussian random variable where the amplitude is Rayleigh distributed and the ...
2k views

### Random signals as power signals

Why are random signals considered as power signals (i.e. signals with infinite energy and finite average power)? Does this make any sense? What does it mean for random signals to have infinite energy ...
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### How to determine if output signals represent the same process with different unknown random inputs

This is a question about how to determine if two different output signals represent the same process with different random inputs. This is related to validating ship motion modeling. When ship motion ...
867 views

### Estimate changing time lag between signals

Suppose I have two timeseries $x_1(t)$ and $x_2(t)$, shown in the image that I drew below. They look almost identical in general form, except some features are shifted slightly in time from one series ...
473 views

### Relation between Covariance matrix & Energy of a random signal

Let's say I have the below random signal: $Y[n] = [y(n), y(n-1), y(n-2), \ldots, y(1)]$ I have two random variables now: The first one $X_1$ which express the maximum eigenvalue of the covariance ...
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### The Standard Deviation of The Derivative of a Signal

Given a signal with zero mean and a standard deviation of 0.1 sampled at 5000 Hz. What would be the Standard Deviation of its 1st, 2nd and 'n' derivative? For instance, let's say we measure the ...
2k views

### Difficulty in understanding ergodicity and ensemble averaging

Literature says that a stationary signal is ergodic, if its ensemble average = time averages. Should it be the statistics computed by time averaging = statistics computed by ensemble averaging?The way ...
I was wondering how to calculate the autocorrelation of a deterministic signal $x(t)$ multiplied by a stochastic process $M(t)$, whose autocorellation $R_M(\tau)$ is known a priori. In my case, $x(t)$ ...