1
vote
1answer
41 views

Show expectation is infinite

Let $X_1,\ldots,X_n$ be independent, identically distributed with expectation 1 and finite variance. Find the limit distribution of $\sqrt{n}(\bar{X}_n^{-1}-1)$. If the random variables are sampled ...
1
vote
1answer
20 views

Conditional probability with balls in urns involving discards

I found this problem in a statistics book, and I'm wondering if my solution is correct. "You and a friend play a game involving 20 balls in an urn, of which 1 is red and 19 are white. The game is ...
0
votes
1answer
15 views

Expected value of Cumulative Hazard

Define $T=\min(T^0,C)$ where $T^0$ is the failure time and $C$ is the censoring time. Define the failure indicator $$\delta = \begin{cases} 1 & \text{if $T^0\leq C$}\\ 0 & \text{if $T^0> ...
0
votes
1answer
16 views

Finding probability given mean and standard deviation

I don't know how to approach this problem: X is normally distributed with a mean of 200 and a standard deviation of 10. Find P(X ≥ 203)
0
votes
0answers
33 views

Joint density of $X_1^2+X_2^2\ \text{and} X_2,\ X_i\sim N(0,1)$

Let $X_1 $ and $X_2$ be iid with a common standard normal distribution. I am looking to find the joint pdf of $Y_1 =X_1^2 +X_2^2$ and $Y_2=X_2$. I know i can use a straight transformation argument but ...
0
votes
1answer
19 views

Expected Residual lifetime

I have a 2 part question. I was able to figure out part 1. I need some help with part 2. I will write out part 1 (and my solution) for completion. Let $T$ be a continuous survival time with survival ...
9
votes
4answers
143 views

Is there a closed form expression for $\int_{- \infty}^\infty \int_{-\infty}^y \frac{1}{2 \pi} e^{-(1/2) ( x^2+y^2 )} \mathrm{d}x\,\mathrm{d}y$?

I have been trying to evaluate the integral: $$\int_{- \infty}^\infty \int_{-\infty}^y \frac{1}{2 \pi} e^{-(1/2) ( x^2+y^2 )}\mathrm {d}x\,\mathrm{d}y$$ I know of course that the integral equals ...
1
vote
2answers
46 views

One Question on Law of Total Probability

Let $(X_n)$ with $n \in \mathbb N_0$ be a discrete martingale. Then I read the following identity which is said to be derived from the law of total probability. $$ \mathbb EX_m = \left( \sum_{n=0}^m ...
1
vote
1answer
35 views

on limits of cumulative density functions

Theorem 1.5.3 of Statistical Inference by Casella and Berger States that the function $F(x)$ is a cdf if and only if the following three conditions hold: 1) $\lim_{x \to - \infty} F(x) = 0 \text{ and ...
2
votes
1answer
39 views

Invariance Properties of Brownian Motion

I am trying to make sense of the Scaling-Invariance and Time-Inversion properties of Brownian motion by producing a sample path. For the record, I am using the following definitions. Let $B(t)$ be the ...
4
votes
2answers
146 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
2
votes
1answer
24 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
2
votes
1answer
56 views

$\mathbb E[\mathbb E(X|Y, Z)|Y]$ or $\mathbb E\{\mathbb E[(X|Y)|Z]\}$?

To begin with, the standard iterated law of probability is as follows. $$ \mathbb E X = \mathbb E [\mathbb E(X|Y)]. (1) $$ I am perfectly happy with $(1)$ and there is also some quite good ...
0
votes
1answer
24 views

Convergence of expecations implies convergence of positive and negative parts?

If we have $E|X_n| \rightarrow E|X|$ does that imply \begin{equation} \lim_{n\rightarrow\infty} E X_n^\pm = X^\pm \end{equation} How about if we only have $EX_n \rightarrow EX$? Is this true in ...
2
votes
2answers
48 views

Sandwiching Limsups & liminfs of expectations

Why is it that if we sandwich a liminf of an expectation between two equal quantities we get that the limit exists? Can we somehow deduce the limsup from that and conclude that it's the same or am I ...
3
votes
2answers
88 views

If $X$ and $Y$ are uniform$(-1,1)$, how can I find the distribution of $W=X^2+Y^2$?

If $Y$ and $X$ are independent uniform (-1,1) random variables, I would like to derive the distribution of $W=X^2+Y^2$. At first I thought that I could use the CDF technique and a geometric ...
0
votes
2answers
43 views

If $X$ has CDF $F$, how can I find the CDF of $U= \max \{0,X \}$?

If $X$ has CDF $F$, how can I find the CDF of $U=\max\{0,X\}$? Obviously the suport of $U$ consists solely of nonnegative values. Am I right then in thinking that for $u=0, F_U (u)=F_X(0)$ and for ...
1
vote
3answers
72 views

Transformation of two independent uniform random variables

Suppose $X,Y \sim \text{Uniform} \left(0,1 \right)$ are independent. Then I need to find the PDF for $W=X/Y$. By the CDF technique this is seen to be : $$F_W( w)=\int_{0}^1 \int_{0}^{wy} ...
1
vote
4answers
48 views

A simple conditional probability problem

Assume that two fair dice are rolled one at a time. Given that the sum of the two numbers that occured was at least $7$, compute the probability that it was equal to $7$. I tried computing the ...
1
vote
2answers
27 views

law of total probability and conditiona probability exercise.

Exercise: Let $X$ be an uniform discrete r.v. with four possible values: 1, 2, 3, 4. Let $Y$ be an exponential variable whose parameter is the value taken by $X$. So, if $X = 3$, $Y$ is Exp (3). ...
0
votes
2answers
22 views

Inverse function of borel sets when function is a constant.

Following a simple proof my professor explained in class I am having problems with a specific step: The proof is of probabilistic nature and we are trying to prove that If $X$ (random variable) is ...
1
vote
2answers
40 views

How to get the number of ways of getting a five card hand that is a straight flush from a standard deck of cards

I do not get the result at this page, ex. 13-7: Suppose that Aces can be either high or low; that is, that {Ace, 2, 3, 4, 5} is a straight, and so is {10, Jack, Queen, King, Ace}. The number of ...
0
votes
1answer
76 views

What is purpose of these paragraph?

The following paragraphs are from my study notes on probability theory. It is a section within the independence discussion. But to me, they seem to appear here out of blue. I do not understand what ...
0
votes
1answer
28 views

Convergence in Probability and in Quadratic Mean for a sequence of random variables

I have been trying to determine whether a sequence of random variables, $X_1,X_2,\ldots,X_n$, such that $$P\left(X_n= \frac{1}{n}\right)=1-\frac{1}{n^2}\\ \text{and}\\ ...
1
vote
1answer
22 views

Bayes with conditional independence

I have a problem that I can't work out I've two conditional independent A,B such as $P(A,B|C) = P(A|C)P(B|C)$ Now I've to find posterior formula for: $P(C | A,B)$, now what I got was pretty ...
2
votes
3answers
66 views

CDF of $max(X,X^2)$

$X$ is uniformly distributed on $[-1,1]$. And $Y=max(X,X^2)$. What is $F_{Y}(t)$ , the CDF of $Y$? My attempt: I tried to graph it, but I think I found wrong. I found the joint pdf $5/6$. Is this ...
1
vote
0answers
51 views

Continuous transition kernel for Markov Chains

I am trying to show that the stationary distribution for a Markov Chain on a continuous state space can be obtained by building a transition density kernel, which obeys the detailed balance rule where ...
1
vote
1answer
73 views

Proving that Markov Chain Monte Carlo converges

I am trying to understand how the very basic Markov Chain Monte Carlo approach works: We try to approximately calculate the expected value $E_{\pi(x)}[X]$ by drawing sequential samples from a Markov ...
0
votes
0answers
35 views

Nonrejection region- equivalently the area determined by confidence interval for $H_0: \hat \beta=\beta^*$

For $H_0: \hat \beta=\beta^*$ I want to prove that the non-rejection region in level of significance approach Will be eqaul to the area determined by upper and lower bounds in confidence interval ...
0
votes
2answers
71 views

What is this distribution???

Let $X_1, X_2, \ldots, X_n$ be a random sample from a population with $E(X_i) = \mu$ for all $i \in \{1,\ldots, n \}$. Define $ Y_i = \begin{cases} 1 & \mbox{ if } X_i < \mu \\ 0 ...
0
votes
0answers
39 views

Please check the question: Compute $EX$

Question: A box contains $10$ balls numbered $1,2,\ldots,10$. A random sample of $7$ balls is selected. $X=$ the smallest of the numbers drawn. Compute $E(X)$ $R(X)= \{1, 2, 3, 4\}$ ...
0
votes
1answer
21 views

Probability question.

How many ways are there to distribute 2 indistinguishable white and 4 indistinquisable black balls into 4 indistinquisable boxes? If the question is asked as "distinct boxes", I can solve. But now, ...
0
votes
0answers
40 views

Conditional probability question

Please check the conditional probabilty question I posted. I solved this. But I am not sure. Thank you:)
0
votes
1answer
46 views

A conditional probabilty question.

Question: $8$ identical balls are randomly distributed into $8$ boxes. Given first box and second box are not both empty, find the probability that first box is not empty? $A:=$ B1 is not ...
1
vote
2answers
64 views

Two probability questions.

I have two questions. (1) solution(1): Sample size $=|S|=12^{20}$ $11^{20}\rightarrow$ guarantee that one box is empty. $10^{20}\rightarrow$ guarantee that two boxes are empty. ...
1
vote
1answer
38 views

Show that the intersection of two probabilities in a certain interval

I am struggeling with the following problem: Suppose that $P(A)= \frac{3}{4}$ and $P(B)= \frac{1}{3}$. Show that $\frac{1}{12} \leq P(A \cap B) \leq \frac{1}{3} $. Basically I try to show this ...
0
votes
1answer
45 views

how many ways are there to distribute 10white and 10black balls into 20 distinct boxes so that at most one box is empty

Question: how many ways are there to distribute 10white and 10black balls into 20 distinct boxes so that at most one box is empty. I understand case-1. But I cannot understand a part of answer ...
0
votes
6answers
111 views

Why is an algebra not a $\sigma$-algebra by induction?

I am studying probability theory by reading Sidney Resnick's "A Probability Path". On page 12 and 13, algebra and $\sigma$-algebra are defined. The only difference between the two is the third ...
0
votes
0answers
20 views

Inverse Fourier Transform of $S_Y(f)$

I have this power spectral density $$ S_Y(f) =\frac{N_0}{4 \pi ^{2} f^{2}}\left [ 1- \cos(2\pi f T) \right ] $$ Can any one help me how to find the Inverse Fourier transform?
6
votes
2answers
294 views

What can I do with measure theory that I can't with probability and statistics

I've studied mathematics and statistics at undergraduate level and am pretty happy with the main concepts. However, I've come across measure theory several times, and I know it is a basis for ...
0
votes
1answer
29 views

PMF of Two Random Variables

X and Y are independent and geometrically distributed random variables with $$ P(X = m) = p(1-p)^{m}, m=0,1,2... $$ $$ P(Y = n) = p(1-p)^{n}, n=0,1,2... $$ To find the probability mass function ...
2
votes
1answer
228 views

Expected value of sample median given the sample mean.

Let $Y$ denote the median and let $\bar{X}$ denote the mean of a random sample of size $n=2k+1$ from a distribution that is $N(\mu,\sigma^2)$. How can I compute $E(Y|\bar{X}=\bar{x})$? Intuitively, ...
0
votes
2answers
39 views

A question on conditional probability

Question: Let $X$ and $Y$ be two random variables. The relationship between the two is as follows. If $Y$ is less than or equal to $1$, then $X$ is equal to $Y$; If $Y$ is more than $1$, then $X$ ...
3
votes
0answers
106 views

Examples of Talagrand's inequality

I am trying to understand Talagrand's inequality and when it gives better results than Markov/Chebyshev/Chernoff. However I find the formal definition hard to understand. Are there any nice simple ...
0
votes
1answer
46 views

Shooting Star (Probability)

Assume that a random experiment consists in centering a telescopic sight on a random star. Let $A_{n}$ ($n \in \mathbb{Z}^{+}$) denote the event that the telescopic sight spots exactly $n$ stars. ...
0
votes
1answer
215 views

Square Root of Random Variables

Question: Suppose that $\displaystyle \frac{2}{\theta_0}\sum_{i=1}^n y_i\sim\displaystyle\chi_{2n}^2$ and $\displaystyle 2\theta_0\sum_{i=1}^n x_i\sim\displaystyle\chi_{2n}^2$. And these two are ...
3
votes
3answers
64 views

$X$ and $1 - X$ are identically distributed

I have a two-fold goal for this question. First, I'm trying my hand at making hypotheses and proving them as far as I can. I want to understand the limits of proof, not just the techniques. Second, ...
2
votes
2answers
97 views

Calculate the following conditional expectation.

Let $\Theta$ and $R$ be two independent random variables, where $R$ has density $f_{R}(r)=re^{-\frac{1}{2}r^2}$ for $r>0$ (zero otherwise) and $\Theta$ is uniform on $(-\pi,\pi)$. Let $X=R ...
2
votes
2answers
186 views

How to handle dependence in a probabilistic ball problem?

I am learning about how to handle dependence when doing probabilistic calculations and I came across this problem I don't understand how to solve. I am given $10$ colored balls painted either red or ...
0
votes
1answer
58 views

Probability of getting same number of black and white pebbles

I am trying to learn some probability and the following problem summarises my current confusion quite well. Any help is gratefully received. I have $n$ pebbles, some of which are black, some white ...