# Questions tagged [random]

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### Determining the average probability of error

I'm trying to solve the following problem: In a binary PAM system, the input to the detector is $$y_m = a_m+n_m+i_m$$ where $a_m = \pm1$ is the desired signal, $n_m$ is a zero-mean Gaussian random ...
40 views

### What types of distributions (except the Gaussian) that follow for random jitter in a correlated signal?

For the simple case, let's consider the correlated signal $s$ with jitter (without noise) as follows: \begin{align} s(t, \theta_i) = \cos(2\pi f (t + \epsilon_t) + \phi + \theta_i), ~~ i=0,1, ... \end{...
114 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 $W=\arctan{\left(\frac{Y}{X}\right)}$. ...
24 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 ...
29 views

### 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 ...
67 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 ...
60 views

2k views

### 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 ...