1
vote
1answer
19 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. ...
0
votes
1answer
18 views

A joint pdf question [on hold]

I need help over a question. I appreciate all helps.Thank you.
-1
votes
0answers
36 views

Derive the expected value of $X^{0.5}$ [on hold]

I am doing a question considering a continuous random variable X and have calculated the expected value and variance from the probability density function given. I am unsure of what the expected value ...
0
votes
1answer
19 views

Conditional Probability of random variables! [on hold]

X, Y, Z are i.i.d continuous random variables. How can I compute (1) P(X>Y|X>Z) (2) P(X>Y|Y>Z) ? It seems to be easy but at the same time, confusing! Help me. Thank you:)
1
vote
0answers
11 views

Bound on difference of two i.i.d. variables [duplicate]

Prove that for every two independent, identically distributed real random varaibles $X,Y$, $$Pr(|X-Y|\leq 2)\leq 3\cdot Pr(|X-Y|\leq 1)$$ [Source: The probabilistic method, Alon and Spencer]
-1
votes
0answers
30 views

Calculating Map estimate [on hold]

Hello everyone I am stuck on this problem: Given N independent measurements from an experiment that generates exponentially distributed random variables: $$f(x)={1\over y}e^{-x\over y}$$ ...
0
votes
2answers
15 views

Dependence of random variables

I need to solve the following problem: Let X be a normal random variable with mean  and standard deviation  and let I, independent of X, be such that P{I = 2} = P{I = -2} = 0.5. Let Y = I X. In ...
0
votes
1answer
21 views

Probability - Random viarbles

A notepad manufacturer requires that 90% of the production is of sufficient quality. To check this, 12 computers are chosen at random every day and tested thoroughly. The day's production is deemed ...
0
votes
1answer
23 views

Infinite boundary for random variables

I have a question Suppose that X and Y are random variables with joint pdf is given by and zero otherwise. I need to find marginal and conditional pdf's.But I don't know how to intagrate over an ...
0
votes
1answer
14 views

Joint distribution of two random variables

I have a question about joint distributions but couldn't find the solution. Suppose that $X$ and $Y$ are two random variables and their joint pdf is given by $$f_{XY}(x,y)=cxy(1-x-y), ...
1
vote
1answer
162 views

“Who's Taller” game with random variables

I have an exercise that I cant get my head around. You play the game of who's taller" in class (of n people). You pick always a random opponent among the people you haven't yet played, compare your ...
0
votes
1answer
17 views

Finding variance and standard deviation of a random variable in an equation

Suppose that X is a random variable with mean 17 and standard deviation 5. Also suppose that Y is a random variable with mean 45 and standard deviation 11. Find the variance and standard deviation of ...
0
votes
1answer
21 views

Finding the mean of a random variable in an equation given standard deviation and mean

Please help! What do I plug into these equations to solve for the mean of Z?? Suppose that X is a random variable with mean 23 and standard deviation 5. Also suppose that Y is a random variable with ...
0
votes
0answers
9 views

Chernoff bound with three possible outputs

A slot machine return requires a player to put in \$1. It returns \$3 with probability $4/25$, returns \$100 with probability $1/200$, and returns nothing otherwise. Using Chernoff bound, what is the ...
1
vote
0answers
15 views

When can I leave the absolute value from Chebyshev's inequality?

I have a positive random variable which distribution is unkown, but its mean is $10$. I have to find an estimation of its variance, given, that $Pr(X\geq9$)=0.9980 I thought of Chebyshev's ...
0
votes
1answer
13 views

Explanation of this situation with two random variables - $X$ conditionally distributed on $N$?

Let $N$ have a Poisson distribution with parameter $\lambda = 1$. Conditional on $N = n$ let $X$ have a uniform distribution over the integers $0, 1, ..., n+1$. What is the marginal distribution of ...
1
vote
1answer
28 views

One-sided variant of Chebyshev inequality

For random variable $X$ with standard deviation $\sigma$, and any $t>0$, show that $$\Pr(X-E[X]\geq t\sigma)\leq\dfrac{1}{1+t^2}.$$ Chebyshev's inequality yields $$\Pr(|X-E[X]|\geq ...
3
votes
3answers
42 views

Ratio of expectations for integer-valued random variable

For a nonnegative integer-valued random variable $Y$ with positive expectation, show that $$\dfrac{E[Y]^2}{E[Y^2]}\leq\Pr[Y\neq 0].$$ I suppose that the probability that $Y=i$ is $x_i$, for ...
0
votes
2answers
12 views

Random variable with finite expectation and unbounded variance

What is an example of a random variable with finite expectation and unbounded variance? I'm thinking about putting $1/n$ probability on each of $n$ equally-spaced points. Then as $n$ approaches ...
0
votes
0answers
29 views

Lack of Memory Property with Random Time

Let Z be an exponential random variable and R an independent nonnegative random variable. Show that Z has the lack of memory property also at the random time R, i.e. P(Z − R > u|Z > R) = P(Z > u). ...
0
votes
2answers
59 views

Are X and Y independent random variables?

$\bullet$ Let $Z$ be uniformly distributed on $[-1,1]$. $\bullet$ $X$ is a random variable such that $X=1$ when $Z>0$ and $X=-1$ otherwise. $\bullet$ $Y$ is a random variable such that ...
0
votes
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 ...
1
vote
1answer
17 views

limsup of sequence of random variables scaled with n^{-1}

I have been stuck on the following problem: Let $X_i$ iid non-negative random variables (not only necessarily integer-valued) and define $ A:=\limsup \frac{X_i}{i}$. Prove that ...
2
votes
1answer
31 views

Find the pdf of $X+Y$ (discrete case)

Given the marginal pdfs: $$p_X(k)= e^{-\lambda}\frac{\lambda^k}{k!}\text{ and }p_Y(k)= e^{-\mu}\frac{\mu^k}{k!},\quad k=0,1,2,\ldots$$ find the pdf of $X+Y$. I know for $W=X+Y$, $p_W(w)= \sum_x ...
1
vote
0answers
19 views

Derivative of stochastic process

I have a set of data of a random process (one sample path). The process is sampled every 10 min and each sample is a 10 min average from a sensor. I can compute the statistics of the random process, ...
0
votes
2answers
31 views

A fair coin is flipped until the first tail appears, in general we win \$ $2^k $. St. Petersburg problem.

For the St.Peterburg problem (Example 3.5.5), find the expected payoff if (a) the amounts won are $c^k$ instead of $2^k$, where $0 < c < 2$. (b) the amounts won are $\log(2^k)$. The original ...
0
votes
1answer
16 views

CdF of Sum of 3 Dependent Random Variables

Given three dependent random variables, $S_1,S_2$ and $S_3$, such that $0<S_1,S_2,S_3<∞$ and assuming known their joint pdf $f_{S_1,S_2,S_3}(s_1,s_2,s_3)$ I would like to find the CdF of their ...
0
votes
1answer
24 views

Computing the probability of two variables of the same sample.

Problem: The mean of a variety of apple is 400g with a standard deviation of 50g. If we choose 2 random apples of this variety, what would be the probability that the first one weights 150g more than ...
1
vote
0answers
37 views

Distribution of family's disposable income, given pdf, find $F(y)$

Problem: In a certain country, the distribution of a family's disposable income, $Y$, is described by the pdf $$f(y) = ye^{-y}$$ for $y \geq 0$. Find the $F(y)$. Attempt: In the book there it ...
3
votes
2answers
40 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 ...
0
votes
3answers
60 views

Why is a probability density function nonnegative?

Let $X$ be a random variable and its density $f$ be defined to be the derivative of its distribution function $F$, i.e. $$\Pr(a< X\le b)=F(b)-F(a)=\int_a^bf(x)\operatorname{dx}$$ Now let ...
0
votes
1answer
23 views

Independence of a couple of random variables [closed]

Let $X,Y,Z$ be three random variables (1-dimentional). Is it true that $(X,Y)$ independent of $Z$ is equivalent to ($X$ independent of $Z$) and ($Y$ independant of $Z$). X and Y are not ...
0
votes
2answers
40 views

Distribution of Gaussian Random Variable [closed]

I am struggling with a small problem here. I have a gaussian random variable: Y~N(1,4). What is the distribution of (Y-1)/2. I have no idea how to proceed with this question. Please help! Best
0
votes
1answer
20 views

Finding the probability using probability distributions.

A contractor is required by a county planning department to submit 1-5 forms in applying for a building permit. Let $Y$ be the number of forms required of the next applicant. The probability that $y$ ...
0
votes
0answers
23 views

Why does marginalizing out normal error just change the variance?

Assume that data follow the following model: $$Y_i \sim Normal(\alpha_0 + \alpha_1\mathbb{I}(F) + \zeta,\hspace{1mm} \sigma^2),$$ where $\zeta \sim Normal(0,\hspace{1mm}\tau^2)$. (In the problem, $F$ ...
2
votes
1answer
59 views

Showing that $(1-u)z^2\leq P(uz\leq |X|)$ when $0<u<1, E(X^2)=1, $ and $0<z<E(|X|)$.

I am trying to show that $$(1-u)z^2\leq P(uz\leq |X|)$$ where $0<u<1, E(X^2)=1, $ and $0<z<E(|X|)$. I've been given a hint to consider Cauchy-Schwarz, however, I don't see where ...
0
votes
1answer
16 views

How can we derive cross covariance $R_\mathrm{xy}(t_1,t_2)=R_\mathrm{yx}^*(t_2,t_1)$?

In random process, cross covariance is nonnegative definite like $$R_\mathrm{xy}(t_1,t_2)=\mathbf{E}(\mathrm{X}(t_1)\mathrm{Y}^*(t_2))=R_\mathrm{yx}^*(t_2,t_1)$$ I'm wondering how it can be derived. ...
2
votes
1answer
17 views

How to find the probability of X=6, where X is the max of 3 6-sided dice

I'm fairly certain the answer is $91/6^3$ (confirming it through a script), but I'm not certain how to solve the problem in a mathematically sound way. I made an attempt that got to the right answer, ...
0
votes
1answer
33 views

The quantile function $F^{-1}(p) = c$ for all $p$ in the interval $(p_0, p_1)$ has the condition that $Pr(X=c) = p_1 - p_0$

Let $X$ be a random variable with c.d.f. $F$ and quantile function $F^{-1}$. Assume the following three conditions: (i) $F^{-1}(p) = c$ for all $p$ in the interval $(p_0, p_1)$, (ii) ...
0
votes
0answers
23 views

Generate two sets of (nonlinearly) dependent random numbers

I would like to find a method to generate two sets of (nonlinearly) dependent random numbers. Solution for linear dependence (that is, correlation). Generate two sets of uncorrelated random numbers ...
0
votes
1answer
65 views

PDF and CDF of probability theory [closed]

The continuous random variable X has pdf $$f(x) =\begin{cases} x/2, \ 0<=x<=2 \\ 0, \ \text{elsewhere} \end{cases} $$ Two independent determinations of X are made. What is the probability ...
0
votes
0answers
14 views

Standard Uniform Distibution with Random Variable

Could someone help explain how to solve the following problem: From my understanding, this problem states that we have a function, Uniform(0, 1), that will generate a random value from 0 to 1 with ...
0
votes
0answers
33 views

what is the probability that there is a string of k consecutive heads?

A coin is flipped n times. Assuming that the flips are independent, with each one coming up heads with probability p, what is the probability that there is a string of k consecutive heads? An answer ...
0
votes
1answer
15 views

prove a theorem about an upper bound of entropy of a random vector

There is a theorem that: if Z is any zero-mean, complex random vector with covariance $E[ZZ^H]=R_z$, then $H(Z)\leq \log|{\pi eR_z}|$, with equality holding if and only if Z has a circularly ...
0
votes
0answers
14 views

Show Y is location-scale if $\sigma > 0$ is unknown

Let X be a random variable having the gamma distribution with shape parameter $\alpha$ and scale parameter $\gamma$, where $\alpha$ is known and $\gamma$ is unknown. Let $Y= \sigma $ log $X$. Show ...
0
votes
1answer
14 views

Number of trials to observe all values of a uniform discrete random variable X with a probability of at least 1-q?

Let X take on p values with equal probability. If n trials are to be conducted to ensure that the probability of not observing any of these p values is less than or equal to q, what is the value of n? ...
3
votes
1answer
29 views

Probability of random assignment to form pairs

So the question goes: I have 100 individuals and 100 different buses, and I randomly assigned each individual to sit on a bus (each bus has equal probability of being selected). How many buses are ...
1
vote
1answer
31 views

What is the variance of this random variable: number of items

Let us assume that we have a capacity $n$ which tends to infinity. We have an infinite number of random variables $X_1, X_2, \dotsc$, where each $X_i$ is independent and identically distributed with ...
0
votes
1answer
60 views

Expected value and Variance calculation

Suppose $f$ is an uniformly distributed random variable with parameters $-1,1$ and $g$ is a Poisson-distributed random variable with parameter $\lambda >0$. We assume that $f$ and $g$ are ...
0
votes
1answer
46 views

Difficult probability problem [closed]

I am stuck in this question $33$ miners are trapped in a mine. There´s an elevator that takes $10$ minutes to go down and another $10$ minutes to pick up a miner and return to the surface. One of ...