3
votes
1answer
52 views

Example of non continuous random variable with continuous CDF

Can someone provide an example of $X$ being a non-continuous random variable with continuous cumulative distribution function? For instance: $X$ is discrete if it takes (at most) a countable number ...
0
votes
0answers
40 views

What did I do wrong when using Jacobian transformation

A device containing two key components fails when, and only when, both components fail. The lifetimes, $T_1$ and $T_2$, of these components are independent with common density function $f (t) = ...
1
vote
1answer
35 views

Question about a change of variable used to compute $E(X)$ from the CDF of $X$

I was studying a proof where the author shows that if the range of x is $\mathbb R_+$ and $F$ is the cumulative distribution function then: $$E[X] = \int_{0}^\infty (1-F(x))dx $$ However on one ...
0
votes
2answers
31 views

probability of a flipped coin

A fair coin is flipped three times. Let $A$ be the event that a head occurs in the first flip and $B$ be the event that exactly one head occurs. a) Find $p(A/B)$ b) Are $A$ and $B$ independent? ...
-1
votes
2answers
31 views

If pages in a book have an iid Poisson number of errors, in 10 pages what is the probability that exactly 3 pages have exactly 1 error?

Suppose the number of spelling error on any given page in particular book can be modeled by a Poisson distribution with $\lambda=2$, and assume that the number of errors on different pages is ...
-1
votes
0answers
39 views

Choosing random marbles until one is divisible by $X$ [on hold]

A box contains twelve marbles on which they are numbered by $1,2,3,...,12$. Now let $X$ represent the number of marbles you must choose with replacement until you obtain one with a number that is ...
0
votes
0answers
25 views

What is the optimal prize for a prize ticket in a raffle [on hold]

What, if any is the optimal price for a prize ticket given the value of a prize? For example if you were to raffle a TV and wanted to cover the cost of the prize? Let say the people were aware of how ...
3
votes
1answer
64 views

Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
0
votes
1answer
22 views

Polynomial joint pdf $f(x,y)$ such that of $f(x) \neq f(y)$

How can I build a polynomial joint pdf $f(x,y)$ for $x \in [x_1, x_2]$ and $y \in [y_1, y_2]$ such that of $f(x) \neq f(y)$ or equivalently, $x$ and $y$ are depended on each other?
5
votes
4answers
128 views

What is the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$ and some of the properties of $E[X \mid Y]$?

I was trying to understand both intuitively and rigorously what the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$. Let me tell you first the things that do make sense to me. $E[X\mid Y=y]$ makes ...
0
votes
0answers
18 views

How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable? [duplicate]

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
0
votes
0answers
21 views

Convergence of Beta Distribution to Bernoulli Distribution [closed]

How will I show that the $$\beta\left(\frac{a}{n} , \frac{b}{n} \right)$$ distribution converges to the $$\operatorname{Bernoulli}\left( \frac{a}{a+b} \right)$$ distribution?
0
votes
1answer
36 views

Show that $Pr[X \gg Y]\approx 1$

Can one show (and how) that $$Pr[X \gg Y]\approx 1$$ for $$X:=\sum_{i=1}^k Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ and $$Y:=\sum_{i=k+1}^{\infty} ...
0
votes
0answers
61 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
1
vote
2answers
73 views

Partial sum of binomial

I 'm trying to figure out a closed form solution for the following summation: $\sum_{j=0}^{\omega} j{n \choose j}p^{j}(1-p)^{n-j}$ where $\omega < n$ Is there any closed form solution?
3
votes
1answer
117 views

Averaging inverse CDFs

Suppose I have two distributions $P$ and $Q$ on the line that admit well defined inverse cumulative distribution functions $F^{-1}_P$ and $F^{-1}_Q$. I define an "average" distribution $A$ as the ...
2
votes
1answer
40 views

Expected Payment under limited policy

The unlimited severity distribution for claim amounts under an auto liability insurance policy is given by the cumulative distribution: $$ F(x) = 1 - 0.8e^{-0.02x}-0.2e^{-0.001x} , x \geq 0$$ ...
0
votes
1answer
35 views

Question about exp. distribution

We know that $X\sim \exp(1),Y\sim \exp(2)$ and they are independent. What is $P(Y>X)$? exp=Exponential... Thank you!
1
vote
2answers
29 views

What is Cumulative Binomial probabilities?

I am new to this so don't know if I am asking the right question as I just read about its usage but didn't know what exactly a Cumulative Binomial probability is. So my question is, What is ...
0
votes
0answers
42 views

Is the following probability distribution stationary/constant

For a conservative system, we know that angular momentum, $l$, and total energy, $E$, are constant, i.e. $\dot{l}=\frac{dl}{dt} = 0$ and $\dot{E}=\frac{dE}{dt} = 0$, where $t$ indicates time. Let ...
1
vote
1answer
39 views

Find Limiting Distribution of $|X_n|$

Let $Z_1,Z_2,...,Z_n,...$ be a sequence of independent standard normal random variables. Let $X_n=\sum^n_{k=1}\frac{Z_k}{\sqrt{k}}$. Does the limiting distribution of $|X_n|$ exists? If yes, find it; ...
-1
votes
2answers
29 views

We are making a Bernoulli experiment…

We are making series of independent Bernoulli experiment with $\frac13$ chance to success. What is the probability that we got success at the first experiment, if we know that we get two successes at ...
0
votes
1answer
32 views

Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
3
votes
1answer
50 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
-1
votes
1answer
23 views

Asymoptotic distribution of identically distributed random variables [closed]

$Y_1, Y_2, ..., Y_N$ are independent and identically distributed random variables with the distribution function $F := F_{Y_1}$ and $F'_n(y) = \frac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{\{Y_i \leq x\}}$ as ...
0
votes
0answers
15 views

Probability density function of an element

How to find the probability density function of $x_m\left(1\le m\le n\right)$ from joint density function, $p_X\left(x_1,x_2,\cdots,x_n\right)$, of $n$ random variables which satisfy following ...
0
votes
0answers
12 views

Error of a Serial Processs

Give random variable X and two processes A, B . Assume that $ Y_{1}, Y_{2}$ are estimated versions of X by using processes A, B respectively, with probability: $P\left \{ \left | X-Y_{1} \right ...
4
votes
2answers
86 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim \mathrm{Bin}(n_i, p_i)$. We know that $$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is ...
0
votes
0answers
16 views
1
vote
1answer
28 views

derivation law from the call option formula

i am reading a article about the option pricing. and i got stuck with some typical statement. $C(K)=\int (x-K)^+\mu(dx)$ is given. here, $\mu$ is implied law of asset price and C(K) is the price ...
2
votes
0answers
22 views

$X$ and $Y$ are i.i.d random variables with finite second moments. $X+Y$ and $X-Y$ are independent, show that $X$ is Gaussian.

$X$ and $Y$ are i.i.d random variables with finite second moments. $X+Y$ and $X-Y$ are independent, show that $X$ is Gaussian. Without loss of generality we may assume that $X$ and $Y$ are ...
3
votes
1answer
70 views

[Probability]need help to understand the following expression

So assume $Y$ and $X$ are exponentially distributed with parameters $y_1$, and $x_1$ respecitively. assume c is a constant. I am having huge trouble to understand the integration of the following ...
0
votes
0answers
18 views

Kullback-Leibler or Jensen-Shannon divergence between two distributions.

i would like to understand well what Kullback-Leibler or Jensen-Shannon divergence between two distributions will tels us about two distribution,for instance let us consider following code ...
1
vote
0answers
15 views

Are functions of independent random variables related to each other by a constant independent

I have $6$ random variables $a,b,c,d,f,g$, each having independent Gaussian distribution. Now I define following three random variables \begin{equation} X = ab - cd\\ Y = cf - ag\\ Z = gd - bf ...
0
votes
1answer
35 views

Measurability and knowledge

there seems to be a subtle relationship between knowledge and measurability. If I have a stochastic process $(X_n)_n$, then for example a stopping time ( other examples would be martingales, ...
1
vote
0answers
30 views

Probability: NEED HELP to Understand with the follow [duplicate]

I need help to understand the probability derviation of a paper. Please help me. For the following, please only treat $|h_{R,B}|^2$ and $|h_{A,R}|^2$ as random variables (other parameters can be ...
2
votes
1answer
22 views

Convergence in total variation

There are the very basic convergence types in probability theory: almost sure, in $L^p$-norm, in measure and in distribution. Besides that there is the concept of convergence in total variation norm. ...
1
vote
1answer
26 views

Product of 2 random variables:domain of integration

I am trying to compute the PDF of the product of two ind. random variables: $Z=XY$, where $0\leq x \leq d$ and $ 0\leq y \leq 1 $. ($0<d<1$) I found this formula : $ f_Z(z) = ...
1
vote
1answer
64 views

Exponential Distribution question

I'm having some trouble understanding the mechanics of how to solve with this distribution. The question: The number of years that a washing machine functions is a random variable whose hazard rate ...
1
vote
0answers
34 views

What is the variance of an arbitrary “good” function of several independent normally distributed random variables

During my studies years ago I came over a formula that states something like if $x_i$ are independent normally distributed variables with variances $\sigma^2_i$ and $f(x_i)$ is differentiable (and ...
0
votes
1answer
35 views

How to prove that convergence in MGF implies Convergence in Distribution?

I know that if the moment generating function of two distribution converges to the same function then the two distribution converges in CDF. But how can we prove this thing explicitly ?
1
vote
1answer
49 views

Extreme Value Theory - Show: Normal to Gumbel

The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory. How can we show that? We have $$P(\max X_i \leq x) = ...
0
votes
0answers
42 views

Total Integral of ordered joint probability is 1 ???

I have four random variables $X_1$, $X_2$, $X_3$ and $X_4$. Their joint dist. is $f(x_1,x_2,x_3,x_4)= \exp(-x_1-x_3)$, where limits are $x_4 = 0$ to $\infty$, $x_3 = x_4$ to $\infty$, $x_2 = x_3-x_4$ ...
0
votes
0answers
16 views

Distribution maximum with small sample related to large sample

Suppose the random variables $X_i$, $i=1,\cdots,n$ and $Y_j$, $j=1,\cdots,m$ all have distribution $F(x)$, with order statistics denoted by $X_{(i)}$ and $Y_{(j)}$. Assuming $n<m$ (e.g. $n=m/100$), ...
1
vote
0answers
20 views

Deriving joint distribution from expectation

Given two random variables $X$ and $Y$ and let $K$ be a constant value. Assume the expectation $\mathbb{E}[X(Y-K)^{+}]$ is given for all possible values of $K\geq 0$. Is there a way to derive the ...
0
votes
0answers
7 views

What is a residual distribution vector

I have a general question about a probability distribution of the following form. Let us assume $\mu_0$ is a propability distribution on the set $A=\{1,2,\dots,n\}$ (in my case its the starting ...
4
votes
2answers
53 views

Homework problem - Ways to test if a density function is cumulative density function

I have a problem that states: Let $F : \mathbb R \to R$ be defined by $$F(x) =\begin{cases}e^{\frac{-1}{x}} &\text{if } x > 0\\ 0&\text{if } x \leq 0\end{cases}$$ Is $F$ a ...
-1
votes
0answers
29 views

Homework help - Random Variable min - can't understand what teacher wants me to do with problem

The problem is: Let X(1), . . . ,X n be independent random variables, with X(i) having an exponential with parameter λ(i) distribution, for any i. Then the distribution of the random variable X = ...
0
votes
2answers
53 views

Homework help finding pdf's of y given pdf's of x - stuck

If anyone can give me the steps on how to find pdf$\,'$s of $y$ given $x$. Let X be a continuous random variable with probability density function given by $$ {\rm f}\left(x\right) ...
0
votes
1answer
32 views

Taking an integration with joint probability integrand.

I encounter a joint probability and I was wondering Am I allowed to rewrite it as an integral by using law of total probability? $$P( \{ f(X,Y) \le g(x,y)\} \cap \{Y>y\} ) = \text{?} = \int_{\xi = ...