All Questions
Tagged with density-function conditional-probability
67 questions
0
votes
0
answers
36
views
Is the following integral of a pdf an identity, i.e. always true?
I am reading a paper and the author starts a proof with this
$$
p(\hat{R}|R) = \int p(\hat{R},\theta|R)d\theta
$$
p is the density function.
Is this something that is always true? Can you help me ...
- 1,275
1
vote
1
answer
358
views
Using distributive law of product operator in conditional probability
Suppose I know the following to be true: $$ f(\mathbf{x} | \mathbf{u}) = \prod_{j=1}^M f(x_j | u_j) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [1] $$
From this statement, is it correct to conclude that the random ...
- 872
0
votes
1
answer
1k
views
How to calculate probability of coming from a dataset for a sampled instance?
Suppose we have two datasets $A$ and $B$ with size $S_A$ and $S_B$. Instances of datasets have multiple features. Consider the specific features $X$ and suppose we are given $PDF$ of this feature in ...
- 51
4
votes
1
answer
802
views
PDF of dependent variables
In my recent question an answer was given, and I am able to compute it myself. Still, I'd like to understand where does that answer come from. Hence, what's the approach to handle dependent variables ...
- 1,264
6
votes
1
answer
3k
views
conditional probability, change of variable and Jacobian
I have a question, and I am guessing that the question arises due to my lack of good understanding in the change of variable technique.
I would like to evaluate $f_X(x)$. When $f_Y(y)$ exists, I can ...
- 965
1
vote
1
answer
95
views
PDF of sum of independent Gaussian variables
I am looking for deriving the pdf of $Z$ where
$Z= (\sum\limits_{i=1}^N a_i X_i +Y_1)^2 + (\sum\limits_{i=1}^N b_i X_i +Y_2)^2$,
where $X_i$ and $Y_i$ are independent, zero mean Gaussian random ...
- 301
4
votes
1
answer
754
views
Convolution of random vectors
Suppose, I have two random vectors $A=[A_1, A_2, \dots A_k]$ and $B=[B_1, B_2, \dots B_m]$. What could be the joint PDF $f_{\mathbf{y}}(y_1,y_2,\dots y_N)$ where
$\mathbf{y}=A \ast B$, here $\ast$ ...
- 301
2
votes
2
answers
154
views
Ordered gamma variables led to an ugly integral
Suppose $X_1,X_2,...X_n$ are i. i. d. random variables with p. d. f.
$$f(x)=xe^{-x}I_{(0,\infty)}\!(x)$$
and let $Y_1,...,Y_n$ be the order statistics for these variables.
a) Find the conditional p. ...
- 425
5
votes
1
answer
316
views
Calculating the marginals given the conditional distributions
Suppose $X$ and $Y$ have conditional distributions given by:
\begin{align}
f(x|y)&\propto ye^{-yx}\;\;\text{for}\;\;0<x<B<\infty\\
g(y|x)&\propto xe^{-xy}\;\;\text{for}\;\;0<y<B&...
- 1,463
3
votes
1
answer
92
views
Joint probability of two correlated RVs
I am trying to get the joint PDF of two RVs $X$ and $Y$ where $aX<Y<bX$, so I am stuck in calculating the probability of
$\mathbb{P}(X<x,Y<y|aX<Y<bX)$
any idea?
- 31
4
votes
2
answers
382
views
Calculation of an "unconstrained" normal distribution (starting from a censored one)
Assume that two r.v. $W$ and $Y|W=w$ with
(1) $W \sim \text{N}(\mu_w,\sigma_w^2)$ (iid)
(2) $Y|W=w \sim \text{N}(w,\sigma_y^2)$ (iid)
Further we only observe $Y$ if $Y$ is less then $W$, i.e.,
(3)...
- 1,113
3
votes
1
answer
133
views
Calculating probability
If $f(x,y)=2x , 0\leq x\leq 1 ,0\leq y\leq 1 $, find $ P(Y < e^{-X} \cap X > Y)$ Given X and Y have joint distribution.
Here is my approach:
$$ P(Y < e^{-X} \cap X > Y) = 1- P(Y > e^{-...
- 851
6
votes
1
answer
391
views
Question about a marginal distribution
If I observe the following:
$X \sim N(\mu_x,\sigma^2_x)$
$Y|X=x \sim N(x,\sigma^2_y)$
My objective is to calculate the marginal distribution of $Y$.
(Since the variance term does not address some ...
- 1,113
11
votes
1
answer
9k
views
Interpretation of conditional density plots
I would like to know how to correctly interpret conditional density plots. I have inserted two below that I created in R with cdplot.
For example, is the ...
- 233
1
vote
1
answer
1k
views
Limits of integration for computing a marginal distribution
I have two functions $f_x$ = $\frac{1}{2}\delta(x-5) + 1/4$ where the 1/4 corresponds to a uniform distribution from 5 to 7. I also have $f_{y|x}$ = $\frac {1}{2}\delta(y-x-4) + 1/4$ which is 1/4 in ...
- 103
4
votes
1
answer
1k
views
Is Perkins et al.'s "skill score" an application of Bayes' theorem?
Perkins et al. (2007) introduce a "skill score" for measuring climate model output against observations. The score basically consists of measuring the overlap between probability density functions of ...
- 5,541
3
votes
2
answers
213
views
Show that, for $t>1$, $P[\frac{Y}{Z}\leq t]=\frac{t-1}{t+1}$
Let the distribution of $X$ be $U(0,1)$. Let U be the length of the shorter of the intervals $(0,X)$ and $(X,1)$; that is, $Z=min(X,1-X)$ and let $Y=1-Z$ be the length of the larger part. Show that, ...
- 2,100