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

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2
votes
1answer
40 views

What does it mean u(dx) in the Fourier transform of a probability measure u?

Let $\mu$ be a probability on $\mathbb{R}^n$ and consider its Fourier transform $\overset{\wedge}{\mu} (u) = \int e^{i (u ,x)} \mu( dx)$, where $(u, x)$ is the scalar product of $u$ and $x$. What ...
0
votes
2answers
17 views

Mean and Variance of Binomial Random Variables?

Let $X$ and $Y$ be independent binomial random variables with parameters $n_1=3$, $n_2=4$, and $p=0.3$ (same $p$ for both), and $Z=X+Y$. What is the mean and variance of $Z$? Based on my current ...
1
vote
3answers
31 views

How did we get $ p\sum_{n=1}^{∞} (1-p)^{n-1}=\frac{p}{1-(1-p)}$

I am not sure how we got below expression.. $$\sum_{n=1}^{∞} P(X=n)= p\sum_{n=1}^{∞} (1-p)^{n-1} = \frac{p}{1-(1-p)} = 1$$ I understand that we calculate expected value for n trials using linearity ...
0
votes
1answer
38 views

Identically Distributed Functions of Random Variables

How can you tell if two functions of random variables are identically distributed. For example, $Y_{1} = 2X_{1}$ and $Y_{2} = X_{1} + X_{2}$ How does one determine whether they are identically ...
-1
votes
0answers
19 views

Independent Random Variables with Uniform Distribution

Let $X_1, X_2,...,X_n$ be independent random variables, each having a uniform distribution over $(0,1)$. Let $Z:=\min(X_1, X_2,...,X_n)$ and $Y:=(X_1, X_2,...,X_n)$. I need to find the cdf and pdf of ...
1
vote
0answers
30 views

If two random variables share the same CDF, do they have identical distributions?

If two random variables share the same cdf, do they have identical distributions? For example, if $X_i$ is a collection of i.i.d random variables from the distribution of $X\sim B(8, 0.4)$, ...
0
votes
1answer
25 views

How to differentiate the standard normal deviation w.r.t. a parameter inside the upper bound

Given that $$N(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-\frac{s^2}{2}}\:ds$$ And that $$d=\frac{1}{\sigma\sqrt{\tau}}\ln\left({\frac{S}{e^{-r\tau}K}}\right)+\sigma\sqrt{\tau}$$ How do I take the ...
0
votes
2answers
28 views

A problem on balls of different colors randomly selected from a box.

I got this problem: Given a 20 balls in a box such that 5 of them are green, 5 are yellow, 5 are red and 5 are blue, We randomly choose ball after ball until we choose the first ball that its color ...
1
vote
2answers
42 views

Prove that $X_1 + … + X_r \sim NB(r,p)$

Let $X_1,...,X_r$ be independent random variables with geometric distribution $X_i \sim Geometric(p)$. Then $$X_1 + ... + X_r \sim NB(r,p)$$ This is what I have tried: $$\begin{eqnarray} ...
0
votes
1answer
21 views

Maximum likelihood estimator for general multinomial

Let $(X_1,\ldots,X_r)\sim\text{multinomial}(n,(p_1,\ldots,p_r))$, where $p_r=1-p_1-\cdots-p_{r-1}$. The random likelihood is $Ap_1^{X_1}\ldots p_r^{X_r}$, for some non-zero $A$. The random ...
0
votes
1answer
43 views

IId random variables from Exponential distribution

If $X_1$ and $X_2$ are iid random variables from the exponential distribution with parameter $\lambda$. I need to find the pdf of $X_1/(X_1 + X_2)$. As of now I have used $X_1=\lambda e^{-\lambda x}$ ...
-1
votes
1answer
42 views

Cdf and Pdf of independent random variables(iid)

Let $X_1, X_2,...,X_n$ be independent random variables, each having a uniform distribution over $(0,1)$. Let $Z:=\min(X_1, X_2,...,X_n)$ and $Y:=(X_1, X_2,...,X_n)$. I need to find the cdf and pdf of ...
1
vote
2answers
29 views

Probability regarding 26 letters and one event before another question

The 26 letters A, B, ... , Z are arrange in a random order. [Equivalently, the letters are selected sequentially at random without replacement.] a) What is the probability that A comes before B in ...
0
votes
1answer
33 views

Pdf of $Y=e^Χ$ for an exponential random variable $X$? [closed]

$X$ is an exponential random variable with $θ=5$. If $Y=e^X$, what is the pdf of Y?
0
votes
0answers
18 views

Distribution of the natural Statistic of an exponential family

I have to show, that the natural statistic of an exponential family also belongs to an exponential family. A distribution belongs to an exponential family, if its density has the following form: $ ...
1
vote
1answer
17 views

Fnd a sequence to be convergence in distribution

Varablies $X_1,\ldots ,X_n$ are independent and $\forall {i\in\{1,\ldots n\}}: X_i \sim \exp(1)$. Find numeric string $a_n$ such that sequence of random variables $$Y_n= \max\{X_1,\ldots, X_n\} - ...
1
vote
0answers
10 views

Poisson Binomial upper tail decreases when all succes probabilities decrease?

I have Poisson Binomial r.v. $K$ with success probabilities $p_{1},\ldots,p_{n}$, where $n$ is odd and $p_{i}\in(0,1)$ for $i=1,\ldots,n$. Let $F=\mathbb{P}[K\geq\frac{n+1}{2}]$, that is, $F$ is the ...
-1
votes
0answers
35 views

What's the distribution of $X = \int^{1}_{0}udB_{u}$? [closed]

Let $X = \int^{1}_{0}udB_{u}$,where $B_{u}$is the Brown motion. What's the distribution of $X$? The Stochastic integral calculate in the sense of Ito.
0
votes
1answer
30 views

Find CDF when knowing PDF, also find E[X]

i lose my login information and have to make new account i apologies for my poor english i am given a problem as such: $$ f(x) = \begin{cases} c(1-x^2), & -1<x<0, \\ c/x^2, & ...
0
votes
0answers
14 views

Is this process deterministic?

Let be $$D_{t}:(\Omega, Q, P)\rightarrow (R,B)$$ $$w\rightarrow U(-1,1) $$ where U is an uniform variable. I have been told in class that $D_{t}$ is deterministic and lineal because if you fix $t$ ...
0
votes
2answers
33 views

Deriving a PDF and finding p [closed]

Suppose that X and Y are independent random variables, X is exponentially distributed with parameter λ=2, and Y is uniformly distributed on (1,3). Find P(Y< X) and derive the pdf of Z=X +Y Any ...
3
votes
3answers
129 views

What is the probability that a Poisson random variable is prime?

Let $X \sim Poisson(\lambda)$, and let $k \in \mathbb{N}$. Consider the quantity $Q(\lambda,k) = P\left( X+k \in Primes\right)$. Obviously $0 < Q(\lambda,k) < 1$. How does $Q(\lambda,k)$ ...
0
votes
1answer
19 views

problem regarding the binomial distribution

I try to solve the following problem: A desktop publisher has to prepare 50 color posters. Each print has a $\frac{1}{6}$ chance of failure (wrong color scheme). How many posters should he print ...
1
vote
1answer
24 views

Probability that this truncated variable is equal to the original

I would like to find this probability $$Pr\{U = X\}$$ where $$U = X \mathbb{1}_{(-\infty,a_n]}(X)$$ $X$ takes on value $2^k$ with probability $\frac{1}{2^kk(k+1)}$ for all $k \geq 1$, and $X = 0$ ...
2
votes
1answer
27 views

Reference for entropy of the binomial distribution?

The Wikipedia page Binomial distribution says that the entropy of the Binomial(n,p) is $\frac{1}{2}\log_2\left(2\pi e n p (1-p)\right) + O\left(\frac{1}{n}\right)$. What is a reference (paper or ...
1
vote
1answer
29 views

Expression for characteristic function of a truncated RV

Let $\langle\Omega,\mathscr{F},\mathbb{P}\rangle$ be a probability space, let $X$ be a random variable defined thereon with density $f$ and $\phi$ be its characteristic function. Then if $A \in ...
1
vote
2answers
29 views

Finding CDF for PDF

¡bom dia! I need to find the CDF for the following: $$ f(x) = \begin{cases} 6(1-x^2), & -1<x<0, \\ 6/x^2, & 1<x<2, \\ 0, & \text{otherwise}. \end{cases} $$ This is more ...
0
votes
0answers
16 views

Expression for joint probability

I have two expressions for $P(X_i|y=0)$ and $P(X_i|y=1)$, where each expression is a multinomial distribution and $y\in \{0,1\}$. I'm interested in finding the joint log likelihood, and thus I'm ...
0
votes
1answer
17 views

Find the point at which the pdf of the chi-squared distribution attains its maximum.

Find the point at which the pdf of the chi-squared distribution attains its maximum. The number of degrees of freedom is $r \ge 2$. My thought was to find the partial derivatives of $x$ and $r$ ...
0
votes
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 ...
1
vote
0answers
28 views

Sum of bernoulli random variables

suppose Z is a random variable which is the sum of some random variables with bernoulli distribution: $Z=Z_1+Z_2+...+Z_m $ , $Z_i \in \{0,1\} ,$ $Pr(Z_i=1)=p=1-1/2^k$ or $1/2^k$ when k is an integer ...
0
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
votes
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, ...
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
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
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 ...
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 ...
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 $\lambda_2$?
1
vote
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 ...
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
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
vote
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 ...
1
vote
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
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) = ...
2
votes
3answers
38 views

Expectation of the function of a random variable

If a random variable $X$ has finite expectation, is the expectation of the function of $X$, e.g. $$f(X)=\exp(X)$$ also finite? How to prove or disprove?
1
vote
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 ...
1
vote
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
vote
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 ...
1
vote
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 ...