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

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1answer
18 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 ...
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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 ...
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1answer
19 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 $P(1.9 \leq \bar{Y}\leq 2.1) >= 0.99$. ...
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0answers
9 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 ...
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2answers
95 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 ...
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0answers
18 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 $E(\epsilon)=0; Var(\epsilon)=\sigma^2$. ...
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0answers
18 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 = ...
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1answer
19 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
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1answer
38 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 ...
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1answer
30 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 ...
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0answers
26 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
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1answer
38 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 ...
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1answer
23 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
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0answers
43 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 ...
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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 ...
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1answer
14 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
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1answer
31 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 ...
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1answer
31 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 $ ...
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1answer
17 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 ...
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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 ...
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2answers
36 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 ...
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3answers
20 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 ...
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1answer
30 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?
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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 $\lambda_2$?
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1answer
34 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 ...
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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 ...
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0answers
25 views

If the probability that a team makes zero hits is $\frac{1}{3}$, what are the changes of getting two or more hits?

Assume that the number of hits, X, that a baseball team makes in a nine-inning game has Poisson distribution. If the probability that a team makes zero hits is $\frac{1}{3}$, what are the changes of ...
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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 ...
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2answers
33 views

Limit of $P(X_n > a_n)$ where $X_n \xrightarrow[n \to \infty]{d} X \sim{N(\mu,\sigma^2)}$ and $a_n\xrightarrow[n \to \infty]{} \infty$

I've been working on following problem and could need some help. Let $X_n$ be a sequence of RV with $$X_n \xrightarrow[n \to \infty]{d} X \sim{N(\mu,\sigma^2)}$$ for some $\mu \in \mathbb{R}$ and ...
0
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1answer
19 views

Probability: determine if $Z[n] = X[n] + Y[n]$ is WSS when $X[n]$ and $Y[n]$ are WSS

Determine if $Z[n] = X[n] + Y[n]$ is WSS when $X[n]$ and $Y[n]$ are WSS and every sample of $X[n]$ is independent of every sample of $Y[n]$. Since both $X[n]$ and $Y[n]$ are WSS, $\mu_X[n] = ...
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1answer
15 views

Probability: IID random process notation question.

I am giving $p_X(x) = \exp(-x)u(x)$. What is $u(x)$? I ask because when I evaluate the probability greater than one, I have $$ \int_1^{\infty}e^{-x}u(x)dx $$ which I know is equal to $e^{-1}$ so why ...
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2answers
69 views

Review Question Help; Discrete Math

Let p be a real number with 0 < p < 1. When and have a child, this child is a boy with probability p and a girl with probability 1 − p, independent of the gender of previous children. Lindsay ...
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1answer
45 views

Indicator random variable review question help

Having a bit of trouble with this review question. A run of ones in a bitstring is a maximal consecutive of ones. For example, the has four runs of ones: , , , and . Let n ≥ 1 be an integer and ...
1
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1answer
46 views

Discrete Math Probability and Random Variable review question

I can't solve this question on my review. If anyone can give me some help to start it, it would be appreciated! Consider an experiment that is successful with probability 0.8. We repeat this ...
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0answers
28 views

Can someone help me figure out where to start?

Let X and Y be two independent random variables. Often one has to consider a new random variable $W = min(X, Y)$ which gives the smaller of the values of $X$ and $Y$ . This exercise studies this ...
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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 ...
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2answers
94 views

Expected value problem: a couple stops having children as soon as they have a child that has the same gender as their first

Givens: $p$ is a real number with $0 < p < 1$ Child is a boy with probability $p$ Child is a girl with probability $1-p$ Anna and Ben stop having children as soon as they have a child that has ...
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2answers
63 views

Proving Expected Value of a Random Variable

Let $r$ and $b$ be positive integers and define $\alpha = \frac{r}{r+b}$. A bowl contains $r$ red balls and $b$ blue balls; thus, $\alpha$ is the fraction of the balls that are red. Consider the ...
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1answer
39 views

A stronger version of Kolmogorov Inequality

I came across this question which says that there is a stronger version of the Kolmogorov Inequality for symmetrically distributed random variables. The question is as follows Let $\xi_1, \ldots, ...
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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 ...
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2answers
52 views

Expected Value Of a Random Variable X

Consider an experiment that is successful with probability $0.8$. We repeat this experiment (independently) until it is successful for the first time. The first $5$ times we do the experiment, we ...
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4answers
85 views

Expected value of a run of a random bitstring

A maximal run of ones in a bitstring is a maximal consecutive substring of ones. For example, the bitstring $1000111110100111$ has four maximal runs of ones: $1, 11111, 1,$ and $111$. ...
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1answer
32 views

Does convergence in distribution of discrete random variables with same finite support imply convergence in probability?

Let ${X_m}\mathop \to \limits^D Y$ and ${\text{supp}}\left( {{X_m}} \right) = {\text{supp}}\left( Y \right) = \left\{ {0,1, \ldots ,n} \right\},\forall m \in \mathbb{N}$. Does ${X_m}\mathop \to ...
1
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1answer
21 views

Given the joint pdf $f_{X,Y}(x,y) = 2e^{-(x+y)}$, $0 \leq x \leq y$, $ y\geq 0$. . Find $P(Y < 1| X = 1)$.

Given the joint pdf $f_{X,Y}(x,y) = 2e^{-(x+y)}$, $0 \leq x \leq y$, $ y\geq 0$. . Find $P(Y < 1| X = 1)$. Attempt: $P(Y < 1| X = 1) \frac{P(Y<1, X = 1)}{P(X = 1)}$ Can someone please ...
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30 views

Applicability of Law of Large numbers for a given sequence of random variables

What are the general methods of showing that law of large numbers doesn't hold for a sequence of random variables?
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1answer
40 views

A confusion on conditional probability

I'm confused on two kinds of conditional probabilities: ${y=x+n}$, where ${x}=\pm1$ with equal probability(0.5). And $n$ is $\cal{N}(0,1)$. Then I know, the conditional probability of ${y}$ ...
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0answers
37 views

Events in the tail $\sigma$-algebra

I am having a little trouble understanding what exactly is the tail $\sigma$-algebra. Just so we are all on the same page, my book defined the tail $\sigma$-algebra like this: Let $X_n$ be a ...
1
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1answer
46 views

Getting a simultaneous first $6$ with two dice

What's the probability that two players that each throw a normal dice get their first six at the same time ? Let $X_k,Y_k$ denote the numbers obtained with the first ($X$) and second ($Y$) dice at ...
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0answers
21 views

Probability of correctly guessing student number with checksum?

At my university, every student has a student number consisting of 8 numbers.The last digit is a checksum so that: the sum of 8 times the first digit, 7 times the second digits, ... and the last digit ...
3
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2answers
50 views

Sum of 'inverse' Normal (1/X) random variables. Equivalent resistance calculation

Consider the case of $N$ resistances $R$ connected in parallel. The equivalent resistance of such a circuit is calculated as follows $$ \frac{1}{R_{eq}} = \underbrace{\frac{1}{R} + \frac{1}{R} + ...