Questions about maps from a probability space to a measure space which are measurable.

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3
votes
1answer
60 views

Difference between two independent binomial random variables with equal success probability

Let $X$ ~ $Bin(n,p)$ and $Y$ ~ $Bin(m,p)$ be two independent random variables. Find the distribution of $Z=X-Y$. see also Difference of two binomial random variables I figured this out: $$ ...
1
vote
1answer
52 views

We roll a standard fair die over and over. What is the expected number of rolls until the first pair of consecutive sixes appears.

During class we split this task into smaller pieces: Let $X$ = r.v. denoting the result of the first roll Let $Y_1$ = r.v. denoting the result of the first roll Let $Y_2$ = r.v. denoting the result ...
3
votes
1answer
39 views

A property about tail equivalent random variables

Let $(X_n)$ and $(Y_n)$ be tail equivalent random variables i.e. $\sum_{i=1}^{\infty}\mathbb P(X_i\neq Y_i)<\infty$ Show that $\sum_{n=1}^{\infty}X_n$ and $\sum_{n=1}^{\infty}Y_n$ converge or ...
0
votes
1answer
43 views

Binomial distribution, when variable isn't x

I've been using the formula $$p(x,N)=\frac{N!}{(\frac{N+x}{2})!(\frac{N-x}{2})!} p^{1/2(N+x)} q^{1/2(N-x)}$$ to determine the probability for a dog who walks in a straight line and can either move ...
-1
votes
2answers
45 views

The product of a normal and Bernoulli variables, independent from each other

Let $X\sim N(0,1)$ and let $Z$ be a random variable independent of $X$ such that: \begin{equation} \Pr(Z=z) = \begin{cases} \frac{1}{2} & \mbox{if $z = -1$ or $z=1$}, \\ \\ 0 & ...
1
vote
2answers
29 views

Coin Tossing Conditional Probability

On a practice test with no available solutions I was asked the following two-part question: 1) If a coin is tossed until three consecutive heads are shown, what is the probability that one tail is ...
1
vote
1answer
25 views

A question about Poisson process such that…

I got the following problem: Suppose that instances of some event occur in accordance with a Poisson process having a rate of 24 instances an hour Suppose we take a time-interval of length 1 hour ...
1
vote
1answer
58 views

Convegence in Probability but not almost surely

Let $ {a_n} $ be a sequence of numbers in $ (0,1) $ such that $ a_n\to0 $, but $\sum_{i=0}^\infty a_n=\infty$. Suppose $X_1, X_2, \ldots$ be independent random variables with ...
3
votes
3answers
52 views

Independence of two normally distributed random variables

Let $X \sim \mathcal{N}(0, 1)$ and $Y$ be a random variable independent of $X$ such that \begin{align*} P(Y=y) = \begin{cases} \frac{1}{2} & y = -1\\ \frac{1}{2} & y = 1\\ 0 & ...
2
votes
1answer
39 views

A question about different pairs that are formed from a set of 16 different poeple such that…

I got the following problem: Given a set of 16 different people. We partition the people into pairs of two. Each pair needs to accomplish a task. And the probability that a pair accomplishes ...
1
vote
1answer
16 views

Suppose a random variable X has mean 0 and moment generating function as follows, find values of a and b

$M_x(t)=a(1+e^{-2t}+e^{-t} +e^t+be^{2t}), -\infty<t<\infty$ Do I take the first derivative of this function? How do I solve for two variables given only one equation? And as a followup ...
0
votes
2answers
33 views

Product of a Continuous and Discrete Distribution

Let $X \sim \mathcal{N}(0, 1)$ and $Y$ be a random variable independent of $X$ such that \begin{align*} P(Y=y) = \begin{cases} \frac{1}{2} & y = -1\\ \frac{1}{2} & y = 1\\ 0 & ...
0
votes
0answers
24 views

Geometric Distributions of Random Variables

Suppose that $X_1, X_2, ...$ are random variables that are independent and geometric, although perhaps with different parameters. Find as many of the following as is feasible: $X_1+X_2, X_1+X_2+X_3$, ...
0
votes
1answer
50 views

Mixed Distributions - Expectation and Variance

A bike has probability of breaking down $p$, on any given day. The repair cost of the bike, whenever it breaks down, is distributed as a Gamma random variable with shape $\alpha$ and rate ...
0
votes
0answers
23 views

Product of two random variables - Resulting Distribution and Correlation?

Let $X \sim \mathcal{N}(0,1)$ and let $Z$ be a random variable independent of $X$ such that \begin{align*} P(Z=z) = \begin{cases}\frac{1}{2} & z=-1\\ \frac{1}{2} & z = 1\\ 0 & ...
2
votes
4answers
49 views

sum and difference between two independent Poisson random variables [closed]

Let $X$ and $Y$ be independent Poisson variables with respective parameters $a$ and $b$. What is the distribution function of $X+Y$? the conditional distribution of $X$, given $X+Y=n$? same, ...
1
vote
1answer
28 views

Expectation values of powers of a Poisson random variable

The expectation values for integer powers of a Poisson random variable $X\sim Poiss(\lambda)$ are well known. I'm interested in the expectation value of $X^\alpha$ for arbitrary rational $\alpha$. ...
-1
votes
1answer
19 views

Find distribution function $F_Y(y)$ of random variable $Y$ [closed]

Tomorrow an midterm exam so I really need your help Let X be a random variable uniformly distributed in $[-1, 4]$. Say $Y = |X|$. Calculate the distribution function $F_y(y)$ of the random ...
0
votes
0answers
18 views

Find Cumulative and the probability density function of Y

Usually I would integrate the function $y=x^2$ from 2 to 1 and to find the probability density function but I need to show it in terms of t. How do I do this? Also is the cumulative distribution = ...
0
votes
0answers
22 views

Necessary Properties of a General Random Variable

Are there any other properties that a random variable X must satisfy besides (1) $X: \Omega \to R $ and (2) the cdf of x be defined for all real x ? Is there anything else that a random variable MUST ...
2
votes
1answer
24 views

Calculating PDF of $Z$ from $X,Y$ when $Z=X+Y$, given the PDFs of $X$ and $Y$

A Student is taking an exam which has two parts, X and Y, with each part given a score from 200 to 800. The students probability distribution for each part is given by $$ f_X(x)= \begin{cases} ...
0
votes
0answers
18 views

If $\mu'$ denotes the pushforward measure, then $\int f\circ X\;d\mu=\int f\;d\mu'$

Let $(\Omega,\mathcal{A})$ and $(\Omega',\mathcal{A}')$ be measure spaces $\mu$ be a measure on $(\Omega,\mathcal{A})$ and $\mu':=\mu\circ X^{-1}$ be the pushforward measure of $\mu$ under $X$ ...
0
votes
1answer
76 views

Unifying the treatment of discrete and continuous random variable

I have been working on the reconciliation of the treatment of discrete and continuous random variable in a measure theoretic sense. But I found myself blocked on fundamental results. We know that If ...
1
vote
1answer
15 views

Application of Borel-Cantelli for sequence of two parameters

Let $(A_{m,\ell})_{\ell \geq 0, m \geq 0}$ be a sequence of events in some probability space. How to show by using Borel Cantelli that, if $$\sum_{\ell \geq 0, m \geq 0} P(A_{m,\ell}) < \infty,$$ ...
0
votes
1answer
28 views

random variables (X,Y) have the following joint PDF

Let the random variable $(X,Y)$ have the following joint PDF $$ f(x,y) = \left\{ \begin{matrix} 2x^{-(2x+y)}, & x>0, y>0\\ 0, & ...
0
votes
1answer
27 views

Bernoulli Distribution (PMF) of random variables X,Y

A fair coin is tossed three times, let X be the number of cases in which the HEAD is obtained, and Y be the absolute value of difference between the number of HEAD and the number of TAIL. Seek the ...
0
votes
2answers
34 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 ...
0
votes
1answer
20 views

Let $Y_{1},Y_{2},…,Y_{n}$ be a normal distribution where $\mu =2$ and $\sigma = 4$. Find $P(1.9 \leq \bar{Y}\leq 2.1) >= 0.99$

Let $Y_{1},Y_{2},...,Y_{n}$ be a random sample from a normal distribution where the mean is $2$ and the variance is $4$. How large must $n$ be in order that $P(1.9 \leq \bar{Y}\leq 2.1) >= 0.99$. ...
0
votes
0answers
11 views

PDF of the logarithm of a chi-squared random variable

Could someone give me a hint, what could be the expression of the PDF of the following random variable Y: Y = a*log(b+X), where a,b are constants and X is a noncentral chi-squared distributed random ...
2
votes
2answers
106 views

Proof of a formula for the expectation of a product of random variables

I want to prove the second task, task b) (see picture below). a) was not hard to show. One question before I start: I am a bit confused about the notation, but $\mathbb 1(t)_{\{Y>t\}}$ is $1$ if ...
0
votes
0answers
19 views

Range of integration for random variables

I have a random variable $Y$ such that $Y=X+\epsilon$ where $X$ is not random variable and takes values in $(0,1)$. $\epsilon$ is random variable with $E(\epsilon)=0; Var(\epsilon)=\sigma^2$. ...
1
vote
0answers
21 views

iid random variables and stopping time

This is Exercise 14.30 from Probability for Statistics and Machine Learning. Let $X_i$ be iid with $E|X| < \infty$, and let $T$ be a stopping time adapted to $\{ X_i \}$. Let $S_n = ...
1
vote
1answer
21 views

If the tubes are shipped in boxes of $1000$, how many wrong-sized tubes per box can doctors expect to find?

The cross-section area of plastic tubing for use in pulmonary resuscitators in normally distributed with $\mu = 12.5mm^{2} $ and $\sigma = 0.2 mm^{2}$ . When the area is less than $12 mm^{2}$ or ...
2
votes
1answer
40 views

In a game, $0.38$ buy hotdogs, how large an order should she place if she wants to have no more that a 20% chance of demand exceeding supply?.

A sell-out crowd of 42,200 is expected at Cleveland's Jacobs Field for next Tuesday's game with the Baltimore Orioles, the last before a long road trip. The ballpark's records from games played either ...
0
votes
1answer
37 views

Markov chain modes of convergence

This is continuation of the question stated here. Let $\left( {{X_\alpha }:\alpha \in A} \right)$ be a finite space Markov chain (discrete or continuous), consisting of only transient and absorbing ...
1
vote
0answers
31 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 ...
2
votes
1answer
41 views

Inequality between variances

Let $X$ be a random variable. Let $a$ and $b$ be two constants such that $a < b$. Define $Z=\min(\max(a,X),b)$. How can we show that: $\text{var}[Z]≤\text{var}[X]$? I think a key step to prove ...
0
votes
1answer
25 views

What is the probability that both the designated captain and the vice captain will not be selected?

If a team of $12$ players is selected randomly from $18$ players, what is the probability that both the designated captain and the vice captain will not be selected. $$1-\frac{12}{18}=\frac1{3}$$ Is ...
2
votes
0answers
49 views

Some properties of a random variable

I have absolutely no idea how to show this: Let $X$ be a random variable whose distribution is not degenerate. By considering the function $F( \theta) = \mathbb{E} U( \theta X)$, $\theta \in ...
0
votes
1answer
21 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 ...
0
votes
1answer
15 views

The time to fix a TV,is an exponential random variable with parameter $\lambda=\frac{1}{2}$.What is the probability that a fix take more than 2 hours?

I got the following question to solve: The time to fix a TV in hours, is an exponential random variable with parameter $\lambda = \frac{1}{2}$. What is the probability that a fix take more than 2 ...
0
votes
1answer
34 views

If the expected value is on the boundary of the range, then the random variable is a.s. constant

Let $X$ be a real-valued random variable on $\Omega$, $I\subseteq\mathbb{R}$ be an interval, $X(\Omega)\subseteq I$ and $E[|X|]<\infty$. Why does $E[X]\in\partial I$ imply that $X=E[X]$ almost ...
0
votes
1answer
34 views

Let $X$ be a normal random variable, with Expected value of 12 and Variance of 4. Find $C$ such that $P(X > C) = 0.1 $

As the title says, I got an exercise I don't know how to approach: Let $X$ be a normal random variable, with Expected value of 12 and Variance of 4. Find $C$ such that $P(X > C) = 0.1 $ ...
0
votes
1answer
19 views

Probability for event to occur exactly $k$ times

The probability to send a computer-word correctly is $0.8$. A computer sends $1000$ words. Let $X$ to be the random variable = "exactly $k$ words sent wrong". What is the distribution of $X$? Is ...
0
votes
1answer
21 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 ...
0
votes
2answers
39 views

A and B flips a coin alternately. A starts. the one how gets `H` first - wins. what is the probability that B wins? [duplicate]

Giving the following question: A and B flips a cion, alternately. A starts. The one who gets H first, wins. Let X be a random variable denotes the ...
1
vote
3answers
22 views

binomial distribution probability - probability for 1 component success out of `n` components

Give the following question: A missile component have a 5% probability to fail. In order to enlarge the probability of success of the missile, we install n ...
1
vote
1answer
35 views

If $X$ and $Y$ are independent random variables, does it follow that $X^2$ and $Y$ are independent? [duplicate]

If $X$ and $Y$ are independent random variables, then can I say that $X^2$ and $Y$ are independent?
-1
votes
2answers
26 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
35 views

Probability that out of their next 100 free throws, they will make between $75$ and $80$, inclusive in basketball game.

State Tech's basketball team, the Fighting Logarithms, have 70% foul-shooting percentage. (a) Write a formula for the exact probability that out of their next 100 free throws, they will make between ...