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

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2
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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
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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
41 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
23 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?
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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 ...
1
vote
1answer
22 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 ...
0
votes
0answers
26 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 ...
1
vote
2answers
43 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 ...
2
votes
2answers
42 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
votes
1answer
21 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] = ...
0
votes
1answer
16 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 ...
1
vote
2answers
77 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 ...
1
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1answer
48 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
vote
1answer
54 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 ...
0
votes
0answers
29 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 ...
1
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2answers
30 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 ...
1
vote
2answers
97 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 ...
1
vote
2answers
65 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 ...
3
votes
1answer
43 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, ...
-1
votes
1answer
34 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 ...
0
votes
2answers
57 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 ...
1
vote
4answers
92 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 $111$. ...
0
votes
1answer
36 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
vote
1answer
22 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 ...
2
votes
1answer
41 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 ${y}$ ...
1
vote
0answers
39 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
vote
1answer
49 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 ...
1
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0answers
22 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
votes
2answers
56 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} + ...
0
votes
1answer
17 views

Continuous random variable transformations

Let $Y$ be a continuous random variable with $f_Y(y)=\frac{1}{2}(1+y).$ Define $W_1=-4Y+7$. Find $f_{W_i}.$ I'm just checking the following solution: $F_{W_1}(w)=P(W\leq w)=P(-4Y+7\leq ...
1
vote
0answers
47 views

Product distribution function of two independent random Variables

Why, if $X $ and $Y $ are two independent'', continuous random variables, described by probability density functions $f_X $ and $f_Y $, then the distribution of $Z = XY$ is $$f_Z(z) = ...
0
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0answers
55 views
2
votes
1answer
50 views

Law of large numbers, problem

I have a specific problem to solve using strong law of large numbers. Let $X_k$ be independent uniform random variables on interval $(0,k)$. Let $Y_n ={1 \over n^2}\sum\limits_{k=1}^n {X_k^3 \over ...
2
votes
1answer
48 views

Proving a certain limit for uniformly integrable random variables

There is an interesting problem that has been resisting my efforts for a while. Assume that $\{X_n: n = 1, 2, \ldots\} $ is a sequence of uniformly integrable random variables. I would like to show ...
0
votes
1answer
37 views

Why is this a martingale.

So I'm looking at this page http://notesofastatisticswatcher.wordpress.com/2012/01/05/a-martingale-that-tends-to-latex-infty-with-probability-1/ where they have this martingale that goes to $-\infty$ ...
3
votes
1answer
54 views

Bound the variance of the product of two random varables.

For two random variables $X$ and $Y$ show that the following inequality holds $$\mathrm{Var}(XY)\leq 2\|Y\|_{\infty}^{2}\mathrm{Var}(X)+2\|X\|_{\infty}^{2}\mathrm{Var}(Y).$$ Well first I tried to ...
2
votes
1answer
46 views

An independent squence of functions that are uniform on $[0,1]$

Suppose that $X$ is uniform in $[0,1]$. Find an infinite sequence of functions $f_{i}$ so that all $f_{i}(X)$ are independent and uniform $[0,1]$. um I'm not really sure how to do this. I'm thinking ...