1
vote
0answers
26 views

What is the variance E[A]^2, statistics? [on hold]

$x(t)= A_i$, for $i \leq t < i + 1$ and $\{i = 0, 1 ,2 ,3,.....\}$. $A_i$ are independent variables, pmf of $P(A_i = 1) = P(A_i = -1) = 1/2$. Find the variance $E[A]^2$. I am so stuck on this ...
0
votes
1answer
21 views

A joint pdf question [on hold]

I need help over a question. I appreciate all helps.Thank you.
0
votes
1answer
25 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
0answers
19 views

Why are Indicator Random Variables better that Random Variables when analyzing algorithms?

I understand the idea behind a random variable and the indicator random variable. BUT my question is why use indicator random variables if we have random variables? How do these indicator random ...
1
vote
0answers
43 views

Joint PDF of Chi-Square & Normal Distribution

Let the independent random variables X1 and X2 be N(0,1) and $\chi^2(r)$, respectively. Let $Y_1$ = $X_1/sqrt(X_2/r)$ and $Y_2$ = $X_2$ a) Find the joint pdf of $Y_1$ and $Y_2$. b) Determine the ...
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
23 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 ...
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 ...
0
votes
2answers
28 views

Question about random variables operations

As we can see in the picture above, what we call random variables looks like much more like a function, in the way that there is an input and then this random variable perfoms a process and gives as ...
1
vote
0answers
20 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
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 ...
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 ...
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
0answers
31 views

Discretization of the standard uniform dist.

I need some help. Sorry for the poor use of LaTEX... a) $U_{n} = \lfloor{nU}\rfloor/n$ prove that $\lim_{n \to +\infty} {U_{n}} = U$ where $U \sim unif[0,1]$ and thus that $lim_{n \to +\infty} ...
0
votes
0answers
27 views

Discrete Random Variable and Its Probability Distribution

EZ Language Center offers a 2-month summer course on three of the most popular and romantic languages aroun the world. French, spanish, and italian. Their database shows that .27, .40 and .33 of their ...
3
votes
0answers
38 views

Distribution of $\frac{X}{|Y|}$, where X and Y are standard normal r.v.'s

Let X and Y be independent standard normal random variables. What is the distribution of $\large \frac{X}{|Y|}$? Attempt: Let $\large U = \frac{X}{|Y|}$ and $ V = |Y|$. This transformation is not ...
0
votes
1answer
58 views

Proving Negative of Standard Normal is Standard Normal

Let X be standard normal random variable $N(1, 0)$ prove that $-X$ is also standard normal. I think I am stuck on a technicality but here is my attempt: Let $Y = -X$ P(Y $\leq$ u) = P($-X$ $\leq$ ...
0
votes
0answers
4 views

A derivation of expected values of difference of random processes

Hope to ask a calculation step from a paper: Let $\mathcal{S}$ be a subset in the Euclidean space $\mathbb{R}^n$. Let $x \in \mathcal{S}$. Let $y \in\mathcal{S}_2^{k-1} = \{y \in \mathbb{R}^k ...
0
votes
0answers
23 views

How to understand a complicated random process

I read a paper and it defines a r.p as following: $x \in \mathbb{R}^n$, $y \in \mathbb{R}^k$ $\{h_i\}_{i=1}^n$ and $\{g_j\}_{j=1}^k$ are two indep. sets of orthonormal r.v.'s Define a r.p: ...
0
votes
1answer
12 views

Proving something is strict stationary

Let $W$ be a uniform distribution on $(0,\pi)$. Let $Z_t=\cos(tw)$. I know that $Z_t$ is a strict stationary but I have no idea how to prove this. Can someone give me some methods?
1
vote
2answers
32 views

Expectation of a function of pairs of random variables

For positive random variables $(X_1, Y_1)$ and $(X_2, Y_2)$, suppose that $(X_1, Y_1)$ and $(X_2, Y_2)$ have the same distribution and (the two pairs) are independent. Also suppose that $E[Y_1|X_1] = ...
1
vote
1answer
43 views

Find the cdf associated with each pdf (NOT transformation)

Find the cdc associated with each pdf: a) f(x) = 3(1-x)^2 , 0 < x < 1 , zero elsewhere b) f(x) = 1/x^2 , -infinity < x < infinity The answers are a) 1-(1-x)^3 , 0 <= x < 1 b) 1 ...
1
vote
2answers
30 views

PMF of X: Number of trials to draw a chip

Let a bowl contain 10 chips of the same size and shape. One and only one of these chips is red. Continue to draw chips from the bowl, one at a time and at random and without replacement, until the red ...
0
votes
2answers
159 views

PMF of number of heads of 4 coin tosses

Let X equal the number of heads in four independent flips of a coin. Using certain assumptions, determine the pmf of X and compute the probability that X is equal to an odd number. I initially ...
2
votes
1answer
29 views

Finding the mean and median of a probability density function

I suspect this is super-easy, but I haven't done any math in about ten years and I'm working with concepts that have been woefully explained... I need to find the mean and median of a continuous ...
3
votes
1answer
27 views

statistics basic question on covariance

anyone would help me in a basic example? a fair coin is tossed, n times. X is the number of Head and Y is the number of Tails. what is the COV(X,Y).
0
votes
3answers
35 views

Variance of the sum of sample means

Let $X$ be a random variable with normal distribution with mean $ \theta$ and variance $ a>0$. Let $ Y $ be a random, variable with normal distribution with mean $\theta$ and variance $b>0$. ...
1
vote
1answer
29 views

Expected length of a random vector

I meet a basic definition about the expected length of a random vector when reading a paper: What is "expected length" How to roughly derive both equations (yellow part) (Is that Gamma ...
3
votes
2answers
85 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
2
votes
2answers
76 views

$X = (X_1, X_2)$ is it not a multivariate random variable?

$X=(X_1,X_2,\ldots, X_P)$ is a $p$-dimensional random variable on $(\Omega, S, P) $ iff $X_i$'s are univariate random variables on the same probability space $(\Omega, S, P)$ ." We all know ...
0
votes
0answers
28 views

probability that a variable, as a function of choice variables, is among the top k out of n when ordered

Suppose $(h_1,h_2,...,h_n)'$ is an $n\times 1$ vector. Let $h_i=g_iX_i$, where $g_i$ is a choice variable which can vary across $i$ and $X_i$ is a random shock with Pareto Type I distribution. ...
-1
votes
1answer
38 views

Odds to guess a 32 byte value [closed]

I have 1,000,000 records, and each is assigned a 32 byte (3.4E+38) random value. What is the likelihood to guess one of the random values? Context This comes up in information security context: ...
1
vote
1answer
45 views

Probability that a random variable is among the top k out of n when ordered

Suppose $X_1,X_2,\ldots,X_n $ are $n$ i.i.d. random variables with a continuous distribution $F(x)$ and density function $f(x)$. What is the probability distribution that any given $X_i$ is among the ...
0
votes
1answer
42 views

$E(Y_i|X_i = 1)$ where $Y_i = X_i + U_i$ with $X_i$ being Bernoulli and $U_i$ being Normal

A network source sends a sequence of zeros and ones, $X_1, X_2, ...$ with $X_i$(iid) Bernoulli with $p = P(X_i = 1), 0 < p < 1$. Due to disturbances the received sequence is $Y_1, Y_2, ...$ ...
0
votes
2answers
45 views

Finding mean from die probability

Example 4.4.5: Suppose that there is a 6-sided die that is weighted in such a way that each time the die is rolled, the probabilities of rolling any of the numbers from 1 to 5 are all equal, ...
1
vote
1answer
106 views

Generating random variates in Excel

I am very confused with a question I have found in relation to Excel. I am hoping someone can help me do this or at-least give me direction in which I can figure out how to do this. So far I don't ...
0
votes
1answer
72 views

Normalizing constants for Extreme value distributions

I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem. Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be ...
2
votes
1answer
44 views

Let X be an exponential random variable with P(X < 1/3) = 0.75. What is E(X)?

Let X be an exponential random variable with P(X < 1/3) = 0.75. What is E(X)? I don't get this. Please help.
0
votes
1answer
33 views

Find the distribution function F(y) [closed]

Can someone show me how to do this problem? Don't know how to format my work here.
1
vote
3answers
65 views

Relationship between Binomial and Bernoulli?

How should I understand the difference or relationship between Binomial and Bernoulli distribution?
2
votes
2answers
38 views

probability, expectation, variance

A 10-digit long number is picked randomly and each digit's pick is independent and has an equal probability of being picked (1/9 because there's digits 1 to 9). Let $X = \#\{\text{missing digits}\}$ ...
1
vote
3answers
56 views

Random Variable Problems?

Can someone show me how to work this out? I can't get the answers in the boxes.
0
votes
1answer
106 views

Finding probability density function of a linear combination of mutually independent normal random variables

I'm finding the probability density function of the random variable U defined in the following manner: $$U=\frac{1}{2}(Y_1+3Y_2)$$ CORRECTION: The line above is supposed to be ...
0
votes
2answers
53 views

Random variable distribution. Reposted

$X$ has distribution $B (30, 0.6)$. Find $P(X \geq 16)$. I know how to find $2$ or $3$ numbers where you use combinations and simply add probabilities for each variable. But this value includes $14$ ...
1
vote
2answers
67 views

How do I transform an r.v. using the floor function? (exponential distribution)

Just had a bash at this question for my Intro to Maths Stats module...I got to the end with a probability density function rather than a probability mass function, namely $f_Y(y) = \lambda a ...
2
votes
1answer
77 views

Variance stabilization for Poisson data

Intro Let $Z > 0$ be a random variable with the mean and variance defined as $\mathbb{E}\{ Z \}$ and $\operatorname{Var}\{ Z \}$, respectively. The variance stabilization transform (VST) $f(z)$ ...
1
vote
0answers
33 views

using Cochran's theorem for sample variance where samples are not identical

IS is possible to use Cochran's theorem to prove that the sample variance of normal variables is chi-square in the case the variables are independent but not identical - they all have the same ...
0
votes
2answers
39 views

What is $\operatorname{Var}[aX+bY+c]$?

I know that $\operatorname{Var}[aX+bY]=\operatorname{Cov}[aX+bY,aX+bY]=a^2\operatorname{Var}[X]+2ab\operatorname{Cov}[X,Y]+b^2\operatorname{Var}[Y]$ (by expanding $(ax+by)(ax+by)$ and letting ...
1
vote
1answer
27 views

Finding efficiency of an estimator for Poisson random variables

$\newcommand{\eff}{\operatorname{eff}}$ I am asked to derive the efficiency of the estimator $\hat{\lambda}_1 = \frac{1}{2}(Y_1+Y_2)$ relative to $\hat{\lambda}_2=\bar{Y}$, where $Y_1,Y_2,\ldots,Y_n$ ...
1
vote
1answer
57 views

$n$-dimensional Gaussian distribution: Iso-density manifold. What else?

Let X be a random variable that follows an $n$-dimensional Gaussian distribution with mean vector $\mu\in\mathbb{R}^n$ and covariance matrix the $n\times n$ symmetric positive matrix $\Sigma$, i.e. ...