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

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1answer
48 views

What can we say about this distribution?

If $A$ and $B$ are independent, with indicator random variables $I_A$ and $I_B$. How can we describe the distribution of $(I_A + I_B)^2$ in terms of $P(A)$ and $P(B)$? I would think its sufficient to ...
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1answer
163 views

Conditional Probability & Stochastic process Example

I'm taking this example from a paper I'm reading. I'm having trouble understanding the logic, and I'm hoping someone can help me. Let $ v_{t}=a \ x_{t}+u_{t} $ where $a$ is some constant and $u$ is ...
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0answers
385 views

Linear combination of Normal Distribution

Suppose X is a N(0, 1) and Y is a N(1, 2). They have a cov[x, y] = 2. What is the distribution of (2*X - 3*Y). I was thinking since they are not independent, any linear combination of them would not ...
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1answer
149 views

Different probability space in the same sample space (probability)

Can someone give me examples to the following problem: Exist 2 different probability space on the same sample space? a probability space is a triple (Ω, σ-algebra , P) P - probability function, Ω - ...
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1answer
151 views

Find formulae for $P(X=x)$ and $P(Y=y)$ in terms of $f$ and $g$

Suppose that random variables $X$ and $Y$, each with a finite number of possible values, have joint probabilities of the form $$P(X=x, Y=y) = f(x)g(y)$$ for some functions $f$ and $g$. How in the ...
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1answer
246 views

finding the prior predictive distribution

I have that Y ~ Bernouill(Pi) and Pi~Beta(a,b), find the prior predictive distribution. I know the pmf of Y is Pi^Y(1-Pi)^1-Y And the pdf of Pi is Pi^(Y+a-1)(1-Pi)^(b-Y) I then multiplied these ...
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2answers
78 views

Why is it possible to have $Y \sim Binomial(100, 0.5)$ with $P(X \ge Y) = 1$

Let $X \sim Binomial(100, 0.9)$. Apparently, it is possible to have another random variable $Y\sim Binomial(100, 0.5)$ for which $P(X \ge Y) = 1$. How is that possible? I don't feel that it is ...
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1answer
143 views

Expectation value

I've been thinking about problem that, conceptually is very simple. I have two random variables $F_{x}$ and $F_{y}$ both of which follow a Gaussian distribution of mean $0$ and variance $1$. Now, I ...
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3answers
44 views

What is the value of A for this F(x) distribution to be valid?

Let $$F(x) = A(1 - \dfrac{1}{e^{x-1}})$$ where $1 < x < \infty$ and $0 \leq F(x) \leq 1$ The question is asking to solve for $A$. My idea is that, $$y = (1 - \dfrac{1}{e^{x-1}}) \in (0, 1)$$ ...
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1answer
88 views

Example of random variables with CDF: $F(x, y) = I_{(x+y\geq 1)}$.

I am trying to find an example of two random variables with joint CDF: $$F(x, y) = \mathcal{P} ( X\leq x, Y \leq y) = I_{(x+y\geq 1)}.$$ I can say that there is one such vector because given CDF is ...
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2answers
100 views

Is an average of power-law curves a power-law curve?

If I look at an exponential function, $p(t) = e^{-\mu t}$ where the parameter $\mu$ varies over a gamma distribution given by the density function $f(\mu) = \frac{1}{\Gamma(a)b^a}\mu^{a-1}e^{-\mu/b}$ ...
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0answers
657 views

Uniform distribution on the unit circle (in the complex plane)

I was trying to prove that for a standard complex Gaussian variable $Z$ it holds that $|Z|^2$ is exponentially distributed with parameter 1, $\frac{Z}{|Z|}$ is uniformly distributed on the unit circle ...
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1answer
62 views

what can we say about $G(.)$?

Given $c \in R$, a deterministic probability density $f(x)$ and its cumulative distribution $F(c)$, what can be said about $G(c)$ where: $G(c)=\int f(x)F\left( x+c\right) dx $ The question ...
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2answers
184 views

A Simple probability question

Suppose you are correct 90% of the time. But in a particular case of 30 questions, you only get 15 correct. What are the odds of that?
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50 views

cumulative probability [closed]

If the total no of trials is 4 and success ratio is 3:1 distributed in two classes.find the binomial probability obtained is 0.125. similarly if total no. of trials is 33 and success ratio is 14:19 ...
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1answer
35 views

Off-lattice Brownian bridges in R^3

Start at a point $(0,0,z_0)$ and take $n$ steps of unit length in a random direction (for each step) in $\mathbb{R}^3$. Let such a walk be valid if the position of the last step, and only the last ...
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1answer
129 views

Making a uniform histogram by random numbers

I have a histogram which shows the frequency of elements in a set. I'd like to add minimum number of elements to the set such that the histogram of the set as defined above becomes fairly uniform. Is ...
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2answers
241 views

Binomial distribution sample vs. population mean

I'm a little confused at this question posed by my prof. He asked us to generate a binomial distribution in R and input whatever variables we wanted. ...
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1answer
110 views

What is this question on random variables asking?

The question states: A random variable $X$ is called symmetric about 0 if for all $x \in \mathbb R$, $\mathbb P(X \geq x) = \mathbb P(X \leq -x)$. Prove that if $X$ is symmetric about 0, ...
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1answer
299 views

What does this distribution mean?

I'm kind of embarrassed that I don't fully get this, but I assume it is because the example is using letters for a sample space instead of numbers. ...
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1answer
122 views

Nash equilibria in games with infinitely many strategies

As a simple example, suppose two players A and B play a game wherein each picks a positive integer, and if they both pick the same integer $N$ then B pays $f(N)$ dollars to A, for some given payoff ...
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1answer
283 views

How do you prove that no matter whether P(A)=1 or 0, A is independent from B

Of course we are assuming that A and B are independent events. I know how to show that if P(A)=1 then P(B)=P(AB), but how do we show that if P(A)=0?
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478 views

Do hashing functions have a probability distribution calculated for their output?

This question might look strange, so I will try to be clear. Consider a hashing function $f : M \mapsto H$ which takes a message with arbitrary length $m \in M$ as input and returns a hash $h \in H$ ...
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1answer
374 views

If A ~ Poisson(a), then what is joint probability mass function of male and female fish?

Suppose that the number of fish in a big sea $A$ adheres to a Poisson distribution. What is joint probability mass function of having $x$ male and $y$ female fish? I thought that I could just divide ...
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1answer
83 views

Sampling and probability generating functions - reference wanted

Suppose I have a huge (effectively infinite) population of widgets. The number of widgets that are broken is given by a random variable $X$, whose probability generating function is $p(z) = E(z^X)$. ...
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3answers
178 views

Where did $\lambda$ come from?

Each second, an ounce of a radioactive substance poissonium emits 5 alpha particles on average. Approximate the probability that exactly 4 alpha particles are emitted by an ounce of poissonium over ...
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1answer
56 views

Why is this a Poisson problem?

Snowflakes are falling at an average rate of 10 flakes per square inch per minute. Calculate the probability that a 2 square-inch region has no snow flakes in a given 5 second time interval. How ...
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2answers
2k views

Please help me solve this exponential distribution problem

Question 1 : The time to service a customer at a bank teller's counter is exponentially distributed with mean of 60 seconds. What is the probability that the three customers in the front of an ...
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1answer
98 views

How can I calculate these normal distribution exercises

I got to calculate the value for the variable $c$ when they give me this intervals but I don't know how to interpret the inequations. a) $P(-c \le z \le c) = .668$. b) $P(c \le |z|) = .016$.
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124 views

Poisson of a Poisson Distribution

Is Po(yPo(x)) a Po(yx) distribution? I got Po(yPo(x)) from moment generating functions but I'm not sure how or if to simplify from there... Thanks.
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0answers
17 views

Summation of hypergeometeric series

I was solving a question on hypergeometric, I have been able to solve up to this part and need a simplification. $$ P(x_1)={_{13}}C_{x_1}/_{52}C_5\sum_{x_2 = 0}^{x_2=5} {_{13}}C_{x_2} \times ...
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0answers
31 views

For $X \sim \mathrm{Binomial}(n,\frac{1}{2})$ does there exist $a,b,c,Y$ s.t. $\Pr[X=x]\Pr[X \le x] \leq a\Pr[Y=bx+c]$?

I need to upper bound some complicated expressions involving binomial distributions: Let $X \sim \mathrm{Binomial}(n,\frac{1}{2})$. I want to find $a,b,c,m$ such that for $Y \sim ...
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0answers
114 views

probability of drawing a deck of cards

Player 1 continuously draws one card each time (without replacement) from a deck of cards and stops when Player 1 gets 3 of Hearts. Player 1 gives the first card he draws to Player 2. From then on, ...
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230 views

probability about drawing cards

Suppose someone continuously draws one card each time from a deck of cards (without replacement), until he/she gets the 3 of Hearts. What is the expected maximum value among all of cards he/she ...
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4answers
2k views

probability - 2 cards with same rank

From a deck of 52 cards,What's the probability that he gets a combination of 2 cards with same rank. Eg: 3♥ 3♠
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1answer
647 views

Finding the joint density of sums of normally distributed random variables.

I am trying to solve the following problem. Let $X_1,X_2,…$ be independent with the common normal density $\eta$, and $S_k= X_1+⋯+X_k$. If $m <n$ find the joint density of $(S_m,S_n)$ and the ...
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1answer
364 views

On the existence of a unique solution

I would like to show that there exists a unique solution in $a$ for the below given equation $$g(a)=\frac{1}{ea^{r}}\Bigg(\int^{y_l}_{-\infty} \ln\left(\frac{1}{ea^{r}}\right)f_0(y)dy$$ ...
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1answer
146 views

How many numbers will I guess correctly?

I play a guessing game. In this game, there are 100 equally-sized, folded-up cards randomly dispersed in a bag. The cards are labeled 1 through 100. I draw out the cards one by one without replacement ...
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1answer
293 views

probability about playing cards

Suppose someone continuously draws one card each time from a deck of cards (without replacement), until he/she gets the 3 of Hearts. What is the expected value of distinct suits (Spades, Hearts, ...
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1answer
67 views

Does this equation have a unique solution?

I have an equation for which I need to show that the solution is unique. Here is my problem: $$\frac{1}{a^{r}}\left(\int^{y_l}_{-\infty} \frac{1}{e}f_0(y)dy+\int^{y_u}_{y_l} ...
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2answers
42 views

How many numbers will I get right?

I play a guessing game. In this game, there are 100 equally-sized, folded-up cards randomly dispersed in a bag. The cards are labeled 1 through 100. I draw out the cards one by one and try to guess ...
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1answer
68 views

Stationary Distribution of T

I'm trying to find the stationary distribution of T, a transition matrix (Markov Chain). After I solve the equations of the matrix, I can't get to their values, does that mean that T doesn't have a ...
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1answer
104 views

Can I combine probabilities from 2 aspects of a related process?

Consider 2 related aspects of a process for prices in a financial market: time & return. Time Say I've identified an exponential distribution that reasonably models the occurrence of the ...
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2answers
258 views

Conditional distribution of continuous random variable $X$ given $|X|$?

If a continuous random variable $X$ has a symmetric distribution around $0$, what is the conditional distribution of $X$ given $|X|$?
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2answers
117 views

Why is the probability of a continuous variable taking a particular value zero? Explain only logically

This question isn't the same as this I don't want a mathematical proof or something of the sort. I want a verbal explanation that intuitively will convince me why this is true. The way I see it, ...
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1answer
877 views

Prove that if $X$ and $Y$ are Normal and independent random variables, $X+Y$ and $X-Y$ are independent

If $X \sim \mathrm{Normal}(\mu,\sigma^2)$ and $Y \sim \mathrm{Normal}(\mu,\sigma^2)$ are independent random variables, how do I prove that $X+Y$ and $X-Y$ are also independent? What happens with the ...
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1answer
139 views

Find the conditional probability that a lightbulb lasts T_2 hours given that it is still alive after T_1 hours.

Time $t \ge 0$ after which a new lightbulb burns out is determined by a distribution that has density $f(t) = \lambda e^{-\lambda t}$..... λ is a positive constant. How do you do conditional ...
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2answers
3k views

How do I tell if this function is a probability density function?

If I have $$f(x)=\sin(x\pi/10)\qquad\text{for}\;0\leq x\leq10.$$ How do I tell if it is a probability density function? And if it isn't how do I normalize it?
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1answer
720 views

How to show that these random variables are pairwise independent?

Given the arrays $C=[C_1,C_2,...,C_N]$ and $S=[S_1,S_2,...,S_N]$ of lengths $N$ with elements that are discrete iid uniform distributed with equal probability (p=1/2) of being $\pm$ 1 Consider the ...
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49 views

How is the hint supposed to be used in this Exponential distribution stimulating exercise?

The following problem is from Jacod's Probability Essentials: Let $$g(y)=-\frac{1}{\lambda}\ln{y}$$ and $$ h(y)=g^{-1}(y)=e^{-\lambda y} $$ The probability density function of $Y$ can be quickly ...