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I don't have an example task (but I think that's not required to answer my question) but I like to know how to understand these notations correctly $$X \sim N(0,1) \;\text{ and } \; Y \sim U(0,1)$$

Obviously, $X,Y$ are random variables, $N$ stands for normal distribution and $U$ stands for uniform distribution. What are the numbers $0,1$ inside those brackets? Are these the intervals where the random variables are distributed in between?

How else can I write $X \sim N(0,1)$ ?

I have checked it on Wikipedia and I see that in general we have $X \sim N(\mu, \sigma^2)$ where $\mu$ is the expected value and $\sigma^2$ the variance of normal distribution. In this case, does that mean we have $\mu=0$ and $\sigma^2=1$?

The density function of normal distribution is $$f(x) = \frac{1}{\sqrt{2\pi \sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$

So for our $X \sim N(0,1)$ we have

$$f(X) = \frac{1}{\sqrt{2\pi}}e^{-\frac{X^2}{2}}$$

I hope you can tell me If I understood this correctly? Because we haven't discussed this part in class and I will write a test soon and I have no contact to the teacher because we are on holidays! :(

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    $\begingroup$ Yeah, this is right! $\endgroup$
    – E-A
    Feb 20, 2018 at 22:44
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    $\begingroup$ Yes, $N(\mu,\sigma^2)$ means normal distribution with mean $\mu$ and variance $\sigma^2$, while I guess $U(0,1)$ means uniform distribution over the interval $(0,1)$. $\endgroup$ Feb 20, 2018 at 22:45

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Let me clarify here: you can think of the parantheses as some input. It turns out, normal distribution can be defined with two parameters, its mean and its variance. By convention, we label that as $N(\mu, \sigma^2)$, where the first parameter is the mean and the second is the variance. (We use N usually for normal distribution if we are talking about distributions)

Similarly, uniform distribution is usually defined with two parameters too: its smallest value and its largest value (You can think of this as its range). As such, we use $U(a,b)$ to denote the uniform distribution between $a$ and $b$. (We use U to label uniform distribution if we are talking about distributions)

Not every distribution takes two parameters; for instance, a popular distribution called Bernoulli distribution (the distribution of a biased coin toss) only takes in one parameter, $p$, which gives the probability of getting a 1 (versus a 0). So, if you see $Y$ ~ Bernoulli($p$), that means Y follows the Bernoulli distribution.

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