# Tag Info

Accepted

### Why is $A\cos(2\pi f_ct)$ a non-stationary process?

A random process is a collection of random variables, one random variable for each time instant. It is best to write the random process as $$\{X(t)\colon -\infty < t < \infty\} \tag{1}$$ where ...
• 20.6k

### If the mean of a random process is constant, does it imply the process is first order stationary?

In the usual sense of the term, first-order stationarity means that the first-order distribution of all the random variables is the same: each $X_t$ has the same CDF, and so the same pdf (or pmf) too ...
• 20.6k
Accepted

### Why is $\sin(t)$ a stationary process?

$\sin(t)$ is no random process, because there's nothing random about it. You could add a random amplitude to get a random process: $$x(t)=A\sin(t)\tag{1}$$ This is a random process because $A$ is a ...
• 90.8k
Accepted

• 90.8k
Accepted

### How to find the output mean and autocorrelation of a non-linear system

The OP's updated working is incorrect. Following up what Hilmar suggested gives \begin{align} Y(t) &= a\left(X(t)\right)^2\\ &= a\left(S(t) + N(t)\right)^2\\ &= a\left(S(t)\right)^2 + 2aS(...
• 20.6k
Accepted

### Why doesn't law of large numbers apply to this stationary time-series?

I believe that you are thinking that each value of $X_t$ is determined by a different realisation of $Y$, which in this example is not true. Suppose that $Y$ is the value that comes out from a dice ...
• 5,030
Accepted

### How to find a variance of sample sequence

$\sigma_r^2=\sigma_y^2\sigma_v^2$ iff 1) $\mathbb{E}\{y\} = \mathbb{E}\{v\}=0$. AND 2) $y$ and $v$ are independent, OR uncorrelated and their squares ($y^2$ and $v^2$) are also uncorrelated. Proof:...
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