Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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1answer
107 views

Distribution $P_X$ and c.d.f $F_X(a)$ of a random variable (defined piecewise)

I am having a bit of trouble computing $P_X$ and $F_X(a)$ for a random variable defined piecewise. Let $\Omega = [0,1]$ equiped with Borel $\sigma-field$ and Lebesgue measure. Ok when it comes to a ...
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1answer
35 views

Superlevel sets of a probability density

Let $f$ be a probability density, and let $\alpha$ be a given value between 0 and 1. There must exist some value $q(\alpha)$ such that the set $\{x:f(x)\geq q(\alpha)\}$ has a mass of $\alpha$. Is ...
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0answers
39 views

How to sum random variables

Let $Z_t = \psi_t |\lambda Z_{(t-1)} + (1-\lambda)\epsilon_t |$ be a random variable where $\epsilon~N(0,1)$ is a Gaussian distributed number, $Z_0 = z_0$ and $\psi \in [-1,1]$ a random variable, ...
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2answers
857 views

Is the sum of two independent geometric random variables with the same success probability a geometric random variable? [duplicate]

Is the sum of two independent geometric random variables with the same success probability parameter a geometric random variable? What is it's distribution? My approach is as follows: $Z=X+Y$ ...
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2answers
42 views

What kind of Distribution is this?

I think the question is asking to find a probability distribution for a discrete random variable. But I am not entirely positive because it asks to determine the sample space, but assign probabilities ...
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0answers
17 views

Parameters of extreme values distribution for a family of distributions

My random variable is defined as $X=\max({x_1,...x_n})$, with $n$ very large. The $x_i$ are iid random variables following a Binomial distribution with with $k$ trials and success-probability $p$. ...
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2answers
51 views

Conditional probability for a RV with exponential distribution

Let X be a positive random variable such that for all $x,y>0$ we have that $$\mathbb{P}[X >x+y | X>x] = \mathbb{P}[X > y]$$ I need to show that X has exponential distribution, i.e, ...
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1answer
211 views

Variance of combined data sets

I was taught that combined variance of two independent variables will be the sum of their variances. However, I wonder if given two sets of independent random variables, say $\{X_m\}$ have mean $A$ ...
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1answer
86 views

Why is the sample Mean a consistent Estimator for the Logistic Distribution?

I think is this a very trivial question, but non the less: How can I show that the $ \hat\theta_n = $ $ \bar x $ is a consistent estimator of $ \theta _0 $. Since $ \theta _o $ is $ \mu $ for the ...
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1answer
58 views

Find the conditional probability density function $f_{y|x}(y,x)$

Assuming $Z$ random variable with $Z=X+Y$, $Z$ depending on $X$ but is independent of $Y$. We know the value of $X$. Also assume we know the joint pdf $f_{x,z}(x,z)$. Find $f_{y|x}(y,x)$. Can you ...
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1answer
104 views

Prove that the absolute value of the difference of two invariant distributions on a Markov chain is invariant

If we have $a(x)$, $b(x)$ which are invariant distributions on a Markov chain $X_n$ with state space $S$, how can I prove that $|a(x)-b(x)|$ is also invariant? I know that I must show that: ...
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1answer
698 views

Convergence types in probability theory : Counterexamples

I know that the following implications are true: $$\text{Almost sure convergence} \Rightarrow \text{ Convergence in probability } \Leftarrow \text{ Convergence in }L^p $$ $$\Downarrow$$ ...
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1answer
58 views

Random variables with equal joint distributions have equal marginal distributions?

We are given two vectors $X=(X_1,X_2, . . . ,X_n)$ and $Y=(Y_1, Y_2, . . . , Y_n)$ with equal joint distributions. Do their marginal distributions $P_{X_i}$ and $ P_{Y_i}$ have to be equal? I have no ...
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0answers
36 views

Chebyshev inequality, lower bound on $P(X \ge 200)$

We throw a die $100$ times. Using Chebyshev inequality find the lower bound on the probability that the sum of spots in these $100$ throws is bigger than $200$. Let $X = $ number of spots after $100$ ...
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1answer
51 views

$X$ and $Y$ are ind. exponentially dist. ran. variables w/para. $\beta_1$ and $\beta_2$. Let $U=X+Y$, verify that $f_u(u)= \int_0^u f_{xy}(u-v,v)dv$.

I am a little lost with transformations with exponential distributions, any help would be much appreciated! The given hint is $0<x<infty$ and $x=u-v$
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1answer
36 views

Selecting a type of distribution for a problem

"Among 30 raffle tickets six are winners. Felicia buys 10 tickets. Find the probability that she got three winners." This problem ask me to first identify a random variable and describe its ...
3
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1answer
23 views

Shock model conditional expectation

Consider a shock model where $A(t) = \sum _{i=1}^{N(t)}A_i e^{-(\alpha-S_i)}$ where $N(t)$ is distributed as a Poisson, $A_i$ is the amplitude of the shock, distributed as $U(0,5)$, and $S{_i}$ is the ...
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2answers
43 views

Support of random variables

I don't need help with the question whether or not the map is invertible, however when it comes to the support of $(Z_1,Z_2)$ being as underlined in green, I would understand if $Z_1, Z_2$ were ...
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2answers
44 views

Setting up the Probability Distribution for Independent Events

Let E, F, G, H be 4 independent events with probability Pr(E) = 0.2, Pr(F) = 0.4, Pr(G) = 0.6 and Pr(H) = 0.8. Let the random variable be the number of these events that occur. a) find the ...
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2answers
50 views

Probability distribution is symmetric at a point, distribution function $F(x) + F(-x) =1$

Show that if the distribution $P_X$ is symmetric at $m \in \mathbb{R}$, and there are no discontinuities of the distribution function, then the distribution function $F_X$ satisfies $F_X(t) + F_X(−t) ...
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0answers
99 views

finite dimensional marginal distribution

I am trying to understand the following theorem: "The distribution of a stochastic process is uniquely determined by the family of all its finite-dimensional marginal distribution (and vice versa)" ...
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1answer
113 views

Even density function, distribution - just checking if my solution is correctW

We are given $f_X$ - density function of a random variable $X$, it is an even function, that is $f_X(t) = f_X(-t), \ t \in \mathbb{R}$ $F_X$ is the distribution function for $X$ Prove that $F_X (t) ...
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2answers
76 views

Computing Expected Value

I am completely lost on what the question is saying. I am trying to think of a Probability Distribution to find the expected value, but I cant understand what is going on. I know how to compute the ...
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2answers
60 views

Definition of Support

I am a bit confused with the definition regarding support. Say the joint density function exists. For the random variables $X$ and $Y$, is the joint support of $X$ and $Y$: the values $X$ and $Y$ can ...
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2answers
395 views

Sum of discrete and continuos random variables with uniform distribution

Could you tell me how to find the distribution of $Z = X+Y$ if $X$ is a random variable with uniform distribution on $[0,1]$ and $Y$ has uniform distribution on $\{-1,0,1\}$? $X$ and $Y$ are ...
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1answer
47 views

Naive Bayes Rule related question

I stucked at the following Baye's rule related probability question. Suppose $X_1,X_2, Z$ be discrete random variables with probability mass function $p(\cdot)$. I was wondering under what ...
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2answers
35 views

Conditional Probability for two independent

Let $X_1$ and $X_2$ be independent geometric random variables having the same parameter $p$. Guess the value of $P\{X_1 = i\mid X_1 + X_2 = n\}$ How do go about making a smart guess for this value? ...
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1answer
115 views

Square of Normal distributed variable?

This is a quick question. If $X\sim\operatorname{Normal}$, is $X^2$ rayleigh distributed? I ask this question is because from wiki, it says $X^2$ is called the Chi-Squared Distribution with a degree ...
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2answers
419 views

Two people meeting, expected time of waiting

$A$ and $B$ are supposed to meet. $A$ arrives in a randomly chosen (uniform distribution) moment between $2$ and $3$ pm. $ B$ arrives at $2$ pm with probability equal to $0,5$ and in a randomly ...
3
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1answer
130 views

Product of three independent random variables

We are given three independent random variables $X, Y, Z$. $X$ has the following Bernoulli distribution: $P(X=1)=\frac{3}{4}, \ P(X=0) = \frac{1}{4}$ $Y$ has a uniform distribution on the interval ...
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2answers
220 views

Why does the log-normal probability density function have that extra “x”?

For a random variable $X \sim N(\mu, \sigma^2)$, the probability density function is $$f(x; \mu, \sigma^2) = \frac{1}{\sqrt{2\pi\sigma^2}} \cdot \exp\left\{ -\frac{(x-\mu)^2}{2\sigma^2} \right\}$$ ...
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2answers
78 views

How to cheat at Dungeons and Dragons Character Generation?

When creating a new character in Dungeons and Dragons, one will typically roll three dice to generate a score for an attribute such as Strength, Agility, Constitution, etc. The probability for any ...
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1answer
13 views

Joint Distribution: Define new random varaible

im working on practice problems from the book and i have come across a question that i do not understand... The joint probability distribution of X and Y is shown in the table: ...
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1answer
40 views

Question Regarding finding the mean and variance of a MGF function?

This question confused me at the end where it says a normal random variable. A breakdown of the answer would be great The Question states: The MGF for the (general) normal distribution is given by ...
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2answers
83 views

Conditional expectation for an exponential random variable

Let $X$ be an exponetial r.v. with parameter $1$, and $c$ be a nonnegative coonstant. I'm trying to guess $E[X|X\land c]$ and $E[X|X\lor c].$ Once I know what they are, I should be able to verify them ...
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0answers
81 views

Joint Probability Density Function of a Sample of A Normal Distribution

Here is the question: Suppose that a random variable is normally distributed with mean μ and variance σ^2 , and we draw a random sample of five observations from this distribution. What is the joint ...
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2answers
204 views

Do moments define distributions?

Do moments define distributions? Suppose I have two random variables $X$ and $Y$. If I know $E[X^k] = E[Y^k]$ for every $k \in \mathbb N$, can I say that $X$ and $Y$ have the same distribution?
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55 views

Random numbers generator

If I know how to generate random numbers from Gaussian distribution (using Box-Muller method), how can I generate random numbers from distribution with pdf ...
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1answer
46 views

Volume and Probability of a region given by a random variable

I am currently reading this paper. It is about nearest neighbors of a query point $X_q\in\mathbb{R}^k$ within a point set $P=\{X_i\mid X_i\in\mathbb{R}^k\}$, where the points have distribution $p(X)$ ...
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0answers
64 views

Poker dice probability of rolling 2 pairs

Poker dice are played by rolling 5 dice. Let A be the event of rolling 2 pairs. (e.g. 1,1,2,2,3.). Find $\mathbb{P}(A)$. So my answer is as follows: $$\mathbb{P}(A) = ...
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1answer
87 views

Limiting Behaviour of Root Mean Square Normal Random Variables - Related to Chi-Squared Distribution

Above is my question. I have done the first part - made hard work of it, albeit, but still, it's done. The next part is where I am stuck. Intuitively, it seems (to me!) like we should have $R_n ...
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3answers
42 views

formula to produce a set of probability distributions for a set of integers between a lower and upper bound with a given mean value

The goal is to establish a set of probabilities to be used to select an integer value where the probability of selecting I is Q, I+1 is R, I+2 is S, ... I+n is Z and such that the integer with the ...
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1answer
29 views

Probability Density of Mapped $y=ax^2$ to Normal Distribution

Let $x$ be a scalar random variable and let $y$ be a scalar quantity such that $y = ax^2$. If $$p(x) = \frac 1 {\sigma\sqrt{2\pi}}\, e^{-x^2/(2\sigma^2)}$$ a) Find the probability density of $y$ ...
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1answer
57 views

Let $X$ be a random variable with continuous CDF $F$. Find CDF for $|X|$ and (if $F\in C^1$) density $f_{|X|}$

Let $X$ be a random variable with continuous distribution function $F$. Find the distribution function $F_{|X|}$ for $|X|$. Suppose that $F\in C^1$ . Find the density for $|X|$. My approach was ...
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0answers
207 views

Relationships between Uniform and Pareto Distributions

If $X$ is uniformly distributed over $(a,b)$ and $Y$ is pareto distributed with parameters $(min,c)$, what is the distribution of Z in the following cases? (a) $Z = X + Y$ (b) $Z = XY$ (c) $Z = ...
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2answers
44 views

Probability of $n$ successes in the first $k$ trials given that there were $n+1$ successes in the total of trials

I'm having trouble with the following problem: A man found that $3$ out of $10$ inspected bottles were defective. What is the probability that the $2$ first defective bottles were found in the first ...
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1answer
63 views

t-student distribution

I've got this problem: Here, if $Z,W$ are independent random variables, and $Z$ has normal standart distribution and $W$ has $\chi^2$ with $n$ degrees of freedom, $T=\frac{Z}{\sqrt{\frac{W}{n}}}$. I ...
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1answer
35 views

Calculating the current age pdf from the lifetime pdf

Let's say I know the form of the lifetime pdf for some object class. If I select an arbitrary object from the class which is still alive and for which I have no ancillary information on its current ...
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2answers
189 views

Sum of two non-identical independent uniform random variables

Let $X\text{~Uniform} [0,1]$ and $Y\text{~Uniform}[0,2]$. Find the distribution of their sum, $Z = X + Y$, using the convolution method. I understand that I have to break this into cases for ...
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0answers
143 views

Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find the joint density.

Let $(X,Y)$ be a uniform random vector on the semicircle of radius $1$. Find $f_{X,Y}(x,y)$ and the marginals $f_X(x)$ and $F_Y(y)$. My attempt: Since the random vector is uniform it will have ...