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Suppose i have two random variables, $X$ and $Y$. WLOG assume $X$ is discrete, and $Y$ is continuous. How do we define the joint distribution between $X$ and $Y$? … EDIT: i should note that, I'm wondering how can it be defined and calculated in terms of PDF and CDF. …

Get the joint distribution first: $$P(X=x|P=p) = \frac{f_{X,P}(x,p)}{f_P(p)} \implies f_{X,P}(x,p) = P(X=x|P=p)f_P(p)$$ $$ f_{X,P}(x,p) = p^x(1-p)^{1-x} \qquad (0\le p \le 1 \text{ and } x=0,1)$$ I am … not sure if this is the correct joint distribution? …

Let $Y_j$ be distributed $\exp(Q_j)$ where $j=1, 2$. If component 1 fails first, then $Y_1$ is observed but $Y_2$ is not ($Y_2$ is censored). … How can I derive the joint pdf of a continuous variable $u = \min(Y_1, Y_2)$ and a discrete variable $V = 1$ if $Y_1 < Y_2$ and $0$ otherwise? …

Let $Y$ be a discrete random variable and let $X$ be an (absolutely) continuous random variable and $f(X, Y)$ a function of these two random variables. Let $P(X, Y)$ be the joint probability measure. … I am now wondering how to properly write the joint expectation value $\mathbb{E}[f(X, Y)]$? …

Yet since $X$ is discrete, I am not sure if I am allowed to move it inside the integral. I understand that there may be many flaws to my approach, so please enlighten me. … Calculate the conditional distribution of $P$ given $X=1$. …

I saw in a textbook that if we have a joint distribution $f(X,Y)$ that is a Gaussian distribution, then we have the mode equal to the mean. … I suppose in a discrete case, it would just be $p_1[X_1, Y_1]' + p_2[X_2,Y_2]'$ (assuming there are only two possible values for $X,Y$) but I don't know how to transform this into continuous case. …

In each experiment, $v_1$ and $v_2$ are corrupted with zero mean Gaussian noise, and then compared to each other, and the maximum of the two is reported. … I observe two samples, $v^o_1$ and $v^{o}_2$, sampled from a Gaussian distribution with mean equal to $v_1$ and $v_2$ (respectively) and standard deviation $\sigma_v$ (equal for the two observations). …

Variable #1 is called "p-value" and has 4 categorical levels. Variable #2 is called "Bayes Factor" and has 7 categorical level. Question Is what I have a contingency table? … In other words, does the table I'm showing below indicate the conditional frequency distribution of one variable given the other or the joint frequency distribution of the two variables together? …

But then the authors write: The joint distribution of the observed data $y$ and the unobserved indicators $z$ conditional on the model parameters can be written $$p(y, z\mid\theta,\lambda) = p(z\ … From the perspective of probability theory we can not just multiply $p(y\mid z,\theta)$ and $p(z\mid\lambda)$, as the former is the pdf of a continous random variable and the latter is the pmf of a discrete …

(a) If $U$ and $V$ are jointly continuous, show that $P(U =V) = 0$. (b) Let $X$ be uniformly distributed on $(0,1)$, and let $Y= X$. Then, $X$ and $Y$ are continuous, and $P(X=Y) = 1$. … (b) $X$ is uniformly distributed on $(0, 1)$, so we have the distribution function $F(x) = x$ for $x \in (0,1)$. Then, we have an integrable function $f(x) = 1$ such that $\int_0^x 1 dx = x = F(x)$. …

Consider the joint probability density function $f_{XY} (x, y) = c(x+y)$ over the range $0 < x < 3$ and $x < y < x + 2$. I know how to do this for a joint discrete distribution, e.g. … y when they're continuous, so I'd really appreciate any help! …

This is a problem from All of Statistics by Wasserman that I have been struggling with for a while. Problem Let $X \sim \text{Uniform}(0,1)$. Let $0<a<b<1$. Let \begin{equation} Y = \begin{cases} 1 & …

If random variables $X,Z$ are jointly distributed, with $f_X(x)$ continuous density of $X$ and $p_Z(z)$ the discrete probability mass at $Z=z$, does Bayes theorem hold in the sense that $$p_{Z|X}(z) = … $$ If not, is there an analogue for such discrete-continuous mixtures? …

All the explanations I come across deal with gaussians and continuous distributions which are too much for me to handle right now. … I want to just understand the simplest case of discrete distributions first, and I'm unable to find the resource online, so here's where I need help. The set up is as follows. …

Compute the Moment Generating Function (MGF) $M_{z}(t)$ of $Z$ Compute the Distribution Function (CDF) $F_{z}$ of $Z$. Is $Z$ an absolutely continuous random variable? … From the theory I have studied, I have some questions regarding how to deal with this kind of exercises: When it comes to sum to sum two independent random variables (discrete or continuous) and then …