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

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1answer
21 views

distribution of (inverse) distribution function

Let $F: \mathbb R \rightarrow [0,1]$ be strictly monotonic increasing distribution function. The random variable $X$ has distribution function $F$ and the random variable $U$ is uniformly distributed ...
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votes
4answers
55 views

Negative binomial distribution - sum of two random variables

Suppose $X, Y$ are independent random variables with $X\sim NB(r,p)$ and $Y\sim NB(s,p)$. Then $$X + Y \sim NB(r+s,p)$$ How do I go about proving this? I'm not sure where to begin, I'd be glad for ...
2
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0answers
23 views

Find constant $c$ for piecewise continuous random variable pdf

apologizes for my poor english I wish to find the pdf for a piece-wise function which is defined as such $$ f(x) = \begin{cases} c(1-x^2), & -1<x<0, \\ c/x^2, & 1<x<2, \\ 0, ...
1
vote
1answer
14 views

BSC channel probability, (binary symmetric channel)

I have question regarding the binary symmetric channel (BSC), which assume each channel use is indepedent (i.e, if you send a '0', then you send '1', each time you send it is indepedent of others). ...
0
votes
1answer
20 views

Conditional probability distribution formulas

I got the following question to solve: The time to fix a TV in hours, is an exponential random variable with parameter λ=$\frac{1}{2}$ What is the probability that a repair will take more ...
1
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3answers
30 views

Probability uniform with transformation

Given $X,Y$ being discrete random variables that are independent and can take on values $[0,1,\dots,N]$ with equal probability, what is the distribution of $\max[X,Y]=Z$? Or any other transformation ...
0
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0answers
16 views

Expected Value and Variance

i'm just breaking my head dealing with this question. suppose we toss a coin 1000 times independently, let X be the number of sequences of 7 times "head". with probability p for head. what is the ...
1
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1answer
21 views

Convert CDF $F$ to $G $ defined by $G(x) = P(X<x)$

Let $X$ be a r.v. whose possible values are $0, 1, 2,... ,$ with CDF $F$. In some countries, rather than using a CDF, the convention is to use the function $G $defined by $G(x) = P(X<x)$ to specify ...
9
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1answer
109 views

Inequality for $N(0,1)$ CDF: $|\log F(v)|\leq |\log F(0)|+|v|+|v|^2$

Suppose that $F$ is the CDF of a standard normal distribution. Hayashi (2000) claims that the following is true $$ |\log F(v)|\leq |\log F(0)|+|v|+|v|^2\quad\text{for all}\quad v. $$ How does ...
0
votes
1answer
16 views

Supremum of sum of exponentially distributed random variables

Let $(X_i)_{i\in\mathbb{N}}$ be independent, exponentially distributed random variables with parameter $\lambda$. Define for $t\gt0$ $N_t:=\sup\{n\in\mathbb{N}:\sum_{k=1}^{n} X_k\le t\}$. Show that ...
3
votes
1answer
75 views

Expected value of the longest run of red balls

Suppose there's an urn containing $r$ red balls and $b$ blue balls. At each trial, I'm drawing a ball at random from the urn, without replacement. Let $R$ denote the event of drawing a red ball, and ...
1
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1answer
1k views

Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
1
vote
1answer
42 views

Is $e^{2(\cos(t)-1)}$ the characteristic function of some random variable?

I am asked to decide whether $$f(t)=e^{2(\cos(t) -1)}$$ is the characteristic function of some random variable. Attempt. I am trying to find directly a possible associated random variable (which ...
0
votes
2answers
23 views

What is the probability $X+Y=0$ for two independent Poisson random variables? [closed]

For two independent Poisson random variables, $X$ and $Y$, with parameters $\lambda_1 > 0$ and $\lambda_2>0$ respectively, how do I find P$\{X+Y=0\}$ in terms of $\lambda_1$ and $\lambda_2$?
2
votes
1answer
173 views

Probability question involving simulations of picking balls from a bag

I’m working on a chemistry problem, which essentially translates to finding the answer to a related probability problem. However, my knowledge in probability is very limited and I'd be grateful if ...
0
votes
0answers
22 views

Observed and expected Fisher information of a Bernoulli Random Variable

If $X$ is a Bernoulli random variable with parameter $p$, the probability mass function is given by $$ f(k) = p^k(1-p)^{1-k} $$ and the loglikelihood, $\ell(p)$, is given by $$ \ell(p) = ...
1
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2answers
36 views

Assumption of a Random error term in a regression

In one of my recent statistics courses, our teacher introduced the linear regression model. The typical $y=\alpha + \beta X + \epsilon$, where $\epsilon$ is a "random" error term. The teacher then ...
0
votes
1answer
40 views

CLT for bounded and dependent sequence

Let $\displaystyle X_1,X_2,...X_n$ be identically distributed such that $\displaystyle Pr\{a \leq X_i\leq b\}=1$ for bounded constants $\displaystyle a,b$. Further Let $\displaystyle ...
0
votes
1answer
24 views

Find constants $A, B$ for cumulative density functions (probability) [closed]

I'm stuck with this question and can't seem to find $A$ & $B$. A continuous random variable $X$, which can only take positive values, has cumulative distribution function of the form $$F(x) = ...
1
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2answers
34 views

A Challenging Probabiliy Question

1 (i) A college is trying to fill one remaining seat in its Masters programme. It judges the merit of any applicant by giving him an entrance test. It is known that there are two interested applicants ...
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1answer
20 views

Probability with qualifications and gender

Qualification Female Male Degree 5 1 None 5 4 School 8 12 Vocation 8 7 I've been going through some ...
1
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3answers
148 views

Probability of Distribution of Apples Question.

I have encountered this question which was actually assigned to a Biology class (the deadline has passed). It seemed simple at first but as more time passes by I realise how difficult it is. This is ...
2
votes
1answer
44 views

Find upper limit of normal distribution integration

Considering the normal distribution with standard deviation equals to 0.9 and mean 2.1: $$ P(X\leq a) = \frac{1}{0.9\sqrt{2\pi}}\int_{-\infty}^{a} e^{-\frac12\frac{(x-2.1)^2}{0.9^2}}\,dx $$ I must ...
3
votes
3answers
51 views

Show that $EX^2 = 2\int_0^\infty x(1-F(x)) dx$ for $X>0$

Let $X>0$ with density $f(x)$, have distribution $F(x)$. Show that $$EX^2 = 2\int_0^\infty x(1-F(x)) dx$$ My attempt: By definition, $F(x)=\int^{x}_{-\infty} f(u) du$. Therefore we can write ...
3
votes
0answers
50 views

An inequality with a characteristic function

It's my first question here, hi. In fact, it derives from my probability theory homework, which appears to be unusually difficult (or I just don't see something): Suppose $X$ is a real valued random ...
1
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2answers
29 views

Probability exponential cdf verification question

If $X_1$, $X_2$, $X_3$ are mutually independent exponential($\lambda$) random variables, what is the $96$th percentile of $3\min\{X_1,X_2,X_3\}$? The answer I got is $\frac{-1.44}{\lambda}$ and I'm ...
0
votes
1answer
19 views

Distribution of difference of points in same tail of normal curve?

If $x$ and $y$ are random values drawn from the part of a normal curve that is greater than a fixed $C$. The distributions of $x$ and $y$ are clearly not normal, but is the distribution of their ...
1
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3answers
27 views

How likely are extreme observations in a probability distribution?

Given a measurement that follows a probability distribution (for the sake of argument, Gaussian) how likely is it that repeated observations on the distribution are an extreme of low or high? I ...
0
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0answers
18 views

Standardize $X,Y$ and Covariance Computes a Correlation, But…?

Well known correlation formula, $$ Corr(X,Y) = \frac{Cov(X,Y)}{SD(X)SD(Y)} $$ It can also be used as $$ Corr(X,Y) = Cov \bigg( \frac{X-E(X)}{SD(X)}, \frac{Y-E(Y)}{SD(Y)} \bigg) $$ That is ...
1
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0answers
23 views

When is the joint density differentiable

My question is the following: given a real random vector $X = (X_1,...,X_k)$ with differentiable marginal densities $f_1,...,f_k$, what extra conditions on the marginals are needed to ensure that the ...
0
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0answers
24 views

probability that at least one observation of a random sample

What is the probability that at least one observation of a random sample of size n=5 from a continuous type distribution exceeds the 90th percentile. Let W~Binomial n=5, P=?. how can i get the p of ...
0
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0answers
11 views

Symmetric interval about the conditional mean

Let $X$ and $Y$ be jointly normally distributed with $\mu_X=20,\mu_Y=40,\sigma_X=3,\sigma_Y=2,\rho=0.6 $. Find a symmetric interval about the conditional mean, so that the probability is 0.90 that Y ...
1
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2answers
27 views

What is the joint distribution of these two obscured exponential ones?

$X$ and $Y$ are independent random variables with $X \sim exponential(\lambda)$ and $Y \sim exponential(\mu)$. It is impossible to obtain direct observations of $X$ and $Y$. Instead, we observe the ...
0
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1answer
51 views

If the difference of two i.i.d. random variables is normal, must the variables themselves be normal?

I previously asked a similar question about the sum of two i.i.d. random variables, thinking the two cases to be equivalent. But I can't see how to apply the proof of that case to this one. It is ...
1
vote
1answer
37 views

PDF of a random vector

I have two independent variables X and Y with CDFs $$F_X(x)=x, \ F_Y(x)=x$$ such that $$0≤|x|≤1$$ I need to find the PDF of a random vector $(min(X,Y),max(X,Y))$. I understand that I should take ...
0
votes
1answer
10 views

pdf (transformations of variables

If X has the pdf $f(x)=\frac13, -1<x<2$, zero elsewhere,find the pdf of $Y=X^4$. here is my solution: The support of $Y$ is $(1,16)$. Now, $P(Y\le y)=P (X\le y^{\frac14})$.. then the cdf of Y is ...
2
votes
1answer
32 views

How to use expectation of a random variable to prove its distribution?

If we know a random variable $X$ satisfies $$E(X^k) = \left\{ \begin{array}{lll} 1 \cdot 3\cdot ...\cdot k & \text{if} & k \text{ is even}\\ 0 & \text{if} & k \text{ is odd}\\ ...
2
votes
2answers
22 views

CDF of independent variables

I am given two independent variables $X$ and $Y$. Where $F_X(x)=F_Y(x)=x^4 \ 0\le x \le 1$. I am looking for CDF of $Z=max(X,Y)$ and $E(Z^2)$ I need some pointers, especially for $Z^2$. Thank you.
0
votes
1answer
264 views

Conditional uniform distribution

I had this question in a quiz, and now that I am reviewing it, I am not sure if why my TA gave me the marks because I am pretty sure I am wrong. Let the r.v. $Y$ follow uniform distribution $U(1,2)$ ...
-1
votes
1answer
26 views

Need to find the distribution density of a random vector [closed]

I have two independent variables $X$ and $Y$ with distribution functions $$f_X(x)=x, \ f_Y(x)=x$$ such that $0\le|x|\le1$. I need to find the distribution density of a random vector ...
1
vote
4answers
49 views

about a difficult and weird Probability question

Let W be the random variable that counts the number of tails before one gets r heads for a coin whose probability of heads is θ. Without using moment generating function, show that the mean and ...
0
votes
1answer
341 views

joint probability distribution of one discrete, one continuous random variable

This is a problem on the joint distribution of a discrete and a continuous random variable. Kitty Oil Co. has decided to drill for oil in 10 different locations; the cost of drilling at each ...
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0answers
29 views

recognize the distribution corresponding to this characteristic function

The characteristic function of a random variable $X$ is given as $$\frac{3+\cos(t)+\cos(2t)}{5}; $$ what is the distribution of $X$? I was thinking of the discrete random variable $X=0,1,2$ with mass ...
1
vote
1answer
20 views

Sum of normally distibuted random variables?

I feel terribly confused over this. If: X~N(μx,σx^2) Y~N(μy,σy^2) both independent random variables Z=X+Y Z~N(μx+μy,σx^2+σy^2) Then why: Xi~N(μi,σi^2) i=1, 2, ..., n X1, X2, ..., Xn are ...
0
votes
1answer
26 views

How to calculate the sum of that probability distribution? [duplicate]

Let $W$ be the random variable that counts the number of tails before one gets $r$ heads for a coin whose probability of heads is $θ$. Without using moment generating function, show that the mean and ...
1
vote
4answers
94 views

Proving two random variables differ with positive probability

EDIT: Despite the help of the posters below, I'm still confused. I'm rephrasing the question slightly. Can someone hep me with rephrased problem: Suppose that $X$ is a random vector and $Y$ a random ...
1
vote
2answers
330 views

PDF and CDF of the division of two Random variables

I have two RVs; their PDF are as the followings: \begin{split} f_{X}(x) = \frac 1 {a} e^{-\frac x {a}}\end{split} and \begin{split} f_{Y}(y) = \frac {y^{L-1}} {b^{L} \Gamma (L)} e^{-\frac y ...
0
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1answer
44 views

conditional probability that 5 red balls were placed in the bowl at random

Place five similar balls (each either red or blue) in a bowl at random as follows: A coin is flipped 5 independent times ad a red ball is placed in the bowl for each head and a blue ball for each ...
0
votes
1answer
1k views

Finding joint cdf and pdf of independent random variables

Let $X$ and $Y$ be independent random variables. Each has an exponential distribution with parameter $\lambda$. Define two new random variables by $W = \min({X,Y}) $ $Z = \max({X,Y})$ Find the ...
0
votes
0answers
19 views

Testing if distribution is geometric and finding is parameter.

I was doubting whether this was better suited on a math page or on a programming page but I thought just start with the fundamentals. I'm working on the following: I have constructed a exponential ...