For questions about the expectation of a random variable: computations, upper/below bounds, etc...

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1answer
16 views

How to find distribution of kth highest of n iid's?

For example, suppose there are five people whose heights are independent and exponentially distributed with mean 5.5 feet. I want to be able to solve problems like 1) Find the expected height of the ...
0
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1answer
19 views

Solving the integral which shows the second moment of subtracting two Beta-distributed Random Variables

Peace be upon you In my project I needed to find the second moment of the subtraction of two Beta-distributed random variables. I have computed it and reached to the following integral which I should ...
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2answers
17 views

Can't see how implication on definition of Martingale was arrived at

A Martingale is a discrete time stochastic process $Z_1, Z_2, ..., Z_n$ for any time $n$ that satisfies $E[|Z_n|] < \infty$ $E[Z_{n+1}| Z_0, Z_1, ..., Z_n] = Z_n$ By the linearity of expectation ...
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1answer
14 views

linear regression, expectation and mean squared error

Let us assume that data is generated according to a true model $$y_i = \beta_{true}x_i + \epsilon_i$$ for $i = 1, ..., n$ Assume that $x_i$ are fixed, and $\epsilon_i$~ N(0, $\sigma^2$) ...
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2answers
39 views

Prove $\mathsf E(N)=\sum_{i=1}^\infty \mathsf P(N\geqslant i)$

We want to prove that $$\mathsf E(N)=\sum_{i=1}^\infty \mathsf P(N\geqslant i)$$ We are given a hint that $$\sum_{i=1}^\infty\mathsf P(N\geqslant i)= \sum_{i=1}^\infty\sum_{k=i}^\infty \mathsf ...
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1answer
23 views

Game of Red balls two drawings are made, which rule would you choose if playing the game, rule A or rule B?

In the game of redball two drawings are made without replacements from a bowl that has four white ping pong balls and two red ping pong balls. The amount won is determined by how many ping-pong balls. ...
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0answers
69 views

Derive the expected value of $X^{0.5}$

I am doing a question considering a continuous random variable $X$ and have calculated $k=1/2$, $E(X)=3/2$ and $V(X)=5/12$ I am unsure of what the expected value of $X^{0.5}$ is. Consider the ...
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5answers
65 views

Guessing on the SATs, is it ever better to leave it blank than to guess?

On most SAT questions, there are 5 answers of which exactly one is correct and exactly four are wrong. If one answers correctly you get $1$ point. If you answer incorrectly, you receive $-\frac14$ ...
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1answer
52 views

How to calculate to the power of 1/n? [closed]

Has anyone figured out how to calculate to the power of $1/n$ ("^1/n") on their financial calculator? I just figured I have been doing it wrong this whole time. Does anyone know the right button to ...
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3answers
45 views

Determining $E|X^{n}|$ for $X \sim N(0,1)$ and $n$ odd.

Let $X \sim N(0,1)$. What is $E|X^{n}|$ for $n \in \mathbb{N}$ odd? Attempt: Since $X = -X$ in distribution, we have that $(-X)^{n} = X^{n} = -X^{n}$ in distribution. Then $$E|X^{n}| = E(X^{n})^{+} ...
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0answers
19 views

Equality of conditional expectations [duplicate]

Let $X,Y$ be integrable random variables such that $\mathbb E(X|\sigma(Y))=Y$ and $\mathbb E (Y|\sigma(X))=X$, where $\sigma(X)$ is the smallest sigma algebra such that $X$ is measurable. Show that ...
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1answer
21 views

How does conditional expectation work when the conditional value isn't given?

For example, if we have the following random variables: $Y\sim Bern\left(\frac{1}{5}\right),$ $ Z = \left\{ \begin{array}{l l} X & \quad \text{Y=1}\\ -4X & \quad \text{Y=0} ...
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2answers
30 views

Expectation of a Poisson Distribution: E[X(X-1)(X-2)(X-3)]

Given $X \sim Poi(\lambda)$, what is the expectation of $\mathbb{E}[X(X-1)(X-2)(X-3)]$? I'm not sure how to approach this. I was thinking of expanding the polynomial, but that led to fairly ugly ...
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2answers
28 views

Evaluating an expectation of the form $\mathbb{E}f(X)$

Suppose I have scalar random variables $$X_1 + \cdots + X_n,$$ defined on some probability space and let $S_n$ be their sum. How can I prove the identity ...
2
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1answer
47 views

Is $\mathbb E(X|\mathcal G)$ an integral of $X$ with respect to some measure?

My instructor defined $\mathbb E(X|\mathcal G)$ in the usual way and mentioned that it can also be characterized as an integral of $X$ with respect to some measure. Similarly to $\mathbb E(X|A)=\int X ...
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0answers
16 views

random variables about birthdays

So lets do away with months/days and assume everyone has a birthday $X_k$ which corresponds to a number from $1$ to $365$, uniformly distributed In a group of $n$ people, let $M=\max (X_k)$ be the ...
1
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1answer
16 views

Stock increase multiplicatively, expected value and variance

Assume that a stock starts with price $1$. Each day, if the stock starts with $q$, then with probability $p$ it increases to $qr$, and with probability $1-p$ it decreases to $q/r$. What is the ...
3
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2answers
49 views

Expected number of cycles in permutation

Consider a random permutation of $1,2,\ldots,n$. What is the expected number of cycles in it? I thought about using linearity of expectation, but here it's not clear how we can break down the main ...
2
votes
1answer
24 views

Lower bound for expectation of squared log?

Is there a (tight) lower bound for $\mathbb{E}[(\log x)^2]$ where $x$ is a non-negative random variable? Jensen's inequality doesn't seem to apply here since the squared of a log isn't convex. Thanks! ...
3
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2answers
455 views

Expected number of times until getting two 6's

What is the expected number of times we need to roll a die until we get two consecutive 6's? By definition, it is $\sum_{i=1}^\infty i\cdot Pr[X=i]$. If we need $i$ rolls, that means the last two ...
0
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1answer
32 views

conditional distribution of X, given Y=y and Z=z, and compute E(X|y,z)

Given: $$f(x,y,z) = \frac23 (x+y+z), \,\,\, 0<x<1,\,\,\, 0<y<1,\,\,0<z<1$$ zero elsewhere. Find the conditional distribution of $X$, given $Y=y$ and $Z=z$, and compute $E(X|y,z)= ...
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2answers
64 views

Conditional expectations of $E(X+Y|z)$

Given: $$f(x,y,z) = \frac23 (x+y+z), \,\,\, 0<x<1,\,\,\, 0<y<1,\,\,0<z<1$$ zero elsewhere.I was instructed to determine the cumulative df of $x,y,z$. Here is my answer $$F (x,y,z) = ...
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0answers
24 views

Evaluate $E(X^2YZ+3XY^4Z^2)= E(X^2YZ)+3E(XY^4Z^2)$

Kindly check my answers please. Given: $$f(x,y,z) = \frac23 (x+y+z), \,\,\, 0<x<1,\,\,\, 0<y<1,\,\,0<z<1$$ zero elsewhere. Is my answer correct? ...
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2answers
40 views

Expectation of maximum of two geometric random variables

Let $X,Y$ be independent geometric random variables, where $X$ has parameter $p$ and $Y$ has parameter $Q$. What is $E[\max(X,Y)]$, and what is $E[X\mid X\leq Y]$? If we follow the definition of ...
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2answers
31 views

Expectation of Transposed random variable

Suppose that the random variable X is uniformly distributed on the interval [0, 1]) (i.e X ∼ U(0, 1)) distribution and suppose that Z = min$(2, 2X^2 + 1)$ . (a) Explain why Z does not have a density ...
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2answers
29 views

Expectation of drawing the second color from an urn

My urn contains 3 red, 2 green and 1 white ball. I pick a ball with replacement until I pick the second color. What is the average number of picks for picking the second color? With the expected ...
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2answers
60 views

Why does $E(XY)^2 \le E(X^2)E(Y^2)$ fail to hold for complex $X, Y$?

Does cauchy schwartz inequality hold only for real vectors? Why is that so?
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0answers
34 views

Does a discrete random variable on a finite sample space always have an expected value?

Let $\Omega=\{A,B,C,D\}$,$\mathcal{F}=\{\emptyset,\{A,B\},\{C,D\},\Omega\}$, $P(\{A,B\})=1/2$, and $P(\{C,D\}) = 1/2$. Now $(\Omega,\mathcal{F},P)$ is a probability space. Let ...
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3answers
83 views

Expected number of good pair of coins [closed]

N coins are being put in a line, each of them is either facing Heads or Tail with equal probability.A pair of indices (i,j) is called good coin pair if coin at index i is facing Heads, and coin at ...
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3answers
19 views

How to compute expected value

How do I solve the expected value of this problem, if I have already calculated the pmf? Let $X$ be a random variable with cumulative distribution function given below: $$F_X(x) = ...
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0answers
15 views

Expectation of truncated negative binomial distribution

This is what I have done, I would like to know if there is a way to simplify it, solve summations,...
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1answer
18 views

For X, Y bounded random variables, $E[X^m Y^n] = E[X^m]E[Y^n]$ for all $m, n \geq 0$ implies X, Y are independent

Assume that X, Y are two bounded random variables. If for any integers $m, n \geq 0$, $E[X^m Y^n] = E[X^m]E[Y^n]$, then X and Y are independent. I've worked out that, for sure, if $E[f(X)g(Y)] = ...
0
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1answer
28 views

The expected revenue problem

Question : A travel agent company organizes a tour with ticket price $\$50$ and the ticket is non-refundable. The company has a bus with $20$ seats. The company knows that the participant might not ...
3
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2answers
42 views

Variation of Chebsyhev: How to prove that?

I have the "job" to prove that for any random variable with standard deviation $\sigma$ and expectation $\mu$ and for any $t>0$ we have $$Pr[X-\mu \geq t \sigma] \leq \frac{1}{1+t^2}.$$ I thought ...
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1answer
39 views

Expectation of truncated Poisson Distribution

I have found that if I have a $Y \sim \mathrm{Poi}(\lambda)$ and $Z=Y \mid Y>0$ then I say $$f_Z(k)=g(k)=Pr(Y=k\mid k>0)=\frac{\lambda^k}{k! (e^\lambda-1)}$$ Now I am trying to compute ...
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0answers
29 views

Expectation of absolute difference of 2 repetition of a continuous random variable.

I was working on a problem below: $X$ is a continuous random variable with cdf $F(x)$. If two values of $X$ are observed say, $X_1$ and $X_2$. Then show that $$E|X_1-X_2| = 4 ...
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2answers
16 views

Marginal P.M.F and Conditional Expectation?

I have a joint density function that is formulated as follows: $$ f_{X,Y}(k,y) = \begin{cases} \frac{\partial{P(X=k, Y\le y)}}{\partial y} = \lambda \frac{(\lambda y)^k}{k!}e^{-2\lambda y} & ...
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1answer
50 views

Prove that $E(\mathbf{u}|\mathbf{X})=\mathbf{0}$ implies $Cov(\mathbf{x},\mathbf{u})=0$

Let \begin{equation} \mathbf{y}=\mathbf{X}\mathbf{\beta}+\mathbf{u} \end{equation} where $\mathbf{y}=\begin{bmatrix}y_1 \\ \vdots \\ y_n\end{bmatrix}$, $\mathbf{X}=\begin{bmatrix}X_{11} & ...
1
vote
1answer
42 views

Tips for evaluating $P(X\gt Y\gt Z)$

Does anyone know of any references for how to evaluate stochastic inequalities? Surprisingly, I can't find any good references for general problems. For example, suppose we have three random ...
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1answer
28 views

Last hit before random time s in Poisson point process - expected value.

I'm stuck computing the expected value of the last hitting time before a time $s$ in the waiting time paradoxon. Suppose we come to a bus stop at a time $s \in \mathbb{R}$, where buses are randomly ...
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1answer
38 views

law of iterated expectations with nested conditioning sets

What I'm given: $\bf{g_i}=\bf{x_i}$$u_i$ where $\mathbf{x_i}$ is a k-dimensional vector and $E(u_i|\mathbf{g_{i-1},...,g_1})=0$ I want to show that $E(u_i|u_{i-1},...,u_1)=0$ My work so far: ...
0
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1answer
26 views

Conditional probability operations

Is the following statement true? $\sum_{j}Pr(a|b=b_j)Pr(b=b_j|c=c^*)=Pr(a|c=c^*)$ I'm using this statement (with a simplification that g takes on a finite number of values) to convince myself that ...
2
votes
1answer
17 views

Expected value plug n chug

If $E[X]=2, E[X^2]=5, E[X^3]=0$, and $E[X^4]=30$, find $E[(X-\pi)^3]$. I keep getting a negative number when I do this out! Please help! Here's my work: $E[(X-\pi)^3]=E[X^3]-3\pi E[X^2]+3\pi^2 E[X] ...
1
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1answer
19 views

Finding expectation from joint PDF

Consider the following joint PDF for random variables $X$ and $Y$: (the height that the shading going up to on the $y$-axis is $0.5$, it just didn't show up for some reason). I'm trying to find ...
2
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1answer
50 views

Elevator Wait Time

Question: A building contains one elevator which can access each floor numbered $0$ through $m>0$. It picks up $n\le m$ passengers in the lobby (floor 0). If it takes the elevator $\alpha$ seconds ...
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0answers
14 views

Second Moment of Intensity Function of Stochastic Process

I'm trying to compute the 1st and 2nd moment of the intensity function of a Hawkes Process. The intensity function is of the form $$\lambda(t)=\lambda_0+\int_{- inf}^tv(t-s)dN_s$$ where $\lambda_0$ ...
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0answers
13 views

How to apply the chain rule inside the expectation operator?

I'm working with a function that involves a term $$ E_t\left[ V(w_{t+1})^{1-\gamma}\right]^{\frac{1-\eta}{1-\gamma}} $$ I need to differentiate this with respect to $w_{t+1}$. How should I apply the ...
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0answers
41 views

On what value of $f(X)$ minimizes $E[(Y-f(X))^2|X]$

$X$ and $Y$ are random variables. The question is: what value of $f(X)$ minimizes $E[(Y-f(X))^2|X]$. I am pretty sure I have found the solution to this problem by writing: $$E[(Y-f(X)-E[X|Y] +E[X|Y] ...
2
votes
1answer
27 views

How to find expectation of geometric distribution 2?

Alec and Bill take alternate turns at kicking a football at a goal, and their probabilities of scoring a goal on each kick are $p_1$ and $p_2$ respectively, independently of previous outcomes. The ...
1
vote
2answers
39 views

How to find expectation of geometric distribution?

On average 1 in 8 people in a particular community is left-handed and the rest are right-handed. A sample of people is chosen at random, one by one, until a left-handed person is obtained. Find the ...