1
vote
1answer
20 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
14 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
49 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
26 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
22 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
29 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
26 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
81 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
73 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
26 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
36 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
39 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
41 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
101 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
60 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
50 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
73 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
51 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
57 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
25 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
26 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
54 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. ...
1
vote
1answer
36 views

Exponential Distribution, Statistics.

I know X has an exponential distribution with parameter $\theta =2$. I was asked to define $Y=lnx$ and determine the suppose of Y and the pdf for Y. Then let $X_1, X_2$ be two independent ...
-2
votes
2answers
58 views

Operations on distributions

Say we have two r.v X and Y which are independent and differently distributed ( for e.g X follows a bell curve and Y follows an exponential distribution with parameter $\lambda > 0$ What are the ...
0
votes
1answer
46 views

Probability Density Function of non decreasing function

Can anyone please help me with this random variable question I've stumbled across. Recall from calculus that a function $h$ is called non-decreasing if $x\leq y$ implies $h(x)\leq h(y)$, for every ...
0
votes
0answers
38 views

Algebra of random variables

I am working on a project involving the implementation of a tool capable of numerically performing some basic algebraic operations (sum,product,inverse..) on independent random variables of different ...
0
votes
2answers
68 views

Expected value of XY

I'm having trouble figuring out the expected value of xy. Random variables x and y are described by the joint PDF: $$f(x,y) = \begin{cases} K & \text{: if } x + y ≤ 1 , x > 0 , y > ...
2
votes
1answer
21 views

Probability and Statistics random independent variables

I can't figure out how to determine if these variables are independent. Any help would be greatly appreciated. Random variables x and y are described by the PDF: $$f(x,y) = \begin{cases} k,& ...
1
vote
2answers
86 views
1
vote
1answer
14 views

Variences and adding them from independent random variables?

If I have 3 random varibles X, Y and Z and X=Y+Z then var(X)=var(Y)+var(Z), but Y=X-Z therefore var(Y)=var(X)+var(Z), it is clear that these two contridict, so what makes one of them right and the ...
4
votes
2answers
90 views

Transformation on a random variable

Can someone please help me with formatting this question? $Y$ is an exponential random variable with parameter $1$. Let $Z=-Y$, what is the pdf of $Z$? Attempt: $$\Pr(-Y< y)=\Pr(Y>-y) ,$$ ...
0
votes
1answer
25 views

Likelihood of Two Binomial Distributed RV's

We are given that Let X1~Bin(n1 = 34, p1) and X2~Bin(n2 = 56, p2) In general, what is the likelihood, L(p1, p2) = f (X1, X2 | p1, p2) for the data X1 and X2 I believe that I am supposed to use a ...
0
votes
1answer
14 views

iid random variables (vectors)

If $(X_{1},Y_{1}), (X_{2}, Y_{2}),...,(X_{n}, Y_{n})$ denote a sequence of iid random variables from $(X,Y)$, can I say that each $X_{i}$ is independent from each $Y_{i}$? Or is it just for the ...
2
votes
2answers
56 views

Does a proportion have to be a rational number?

Does a proportion have to be a rational number? For example, Assume we have a square with side $2$ units. We are throwing a circle of radius $1$ unit over the square. Let $X$ be the area of the ...
1
vote
1answer
19 views

Confusion about random variables and convergence in probabilty and distribution

I'm studying statistical analysis and there's something fundamental I'm missing about random variables and how they are used in defining convergence in probability or distribution: In my syllabus ...
0
votes
0answers
29 views

linear system output when input is a Gaussian process?

Rectently, I read a technical book that says:" the linear transform of a Guassian process is also a Guassian process. i.e. for continuous time case: $$ x(t)*h(t)=y(t)$$ the input $x(t)$ is a ...
0
votes
0answers
34 views

Order statistics of random variables

Let $\{I_1, I_2, \dotsc, I_N\}$ be $N$ i.i.d random variables. I know that the smallest orders statistics and the largest one are defined respectively as follow: $$I_{(1)}=\min(\{I_1, I_2, \dotsc, ...
1
vote
1answer
55 views

Find the probability generating function

I have an exercise of this type that I just can not solve "Are $x$ and $y$ be independent random variables, $X$-Poisson($a$), $Y$-Poisson($b$). Find the probability generating function of the random ...
1
vote
1answer
46 views

How do we square a random variable?

How do we square a random variable? For example, Let $Y=X^2$. $$f_X(x)={\frac{1}{\sqrt{2\pi}}} \cdot e^{\tfrac{-x^2}{2}}$$ How do we derive $f_Y(y)$? Thanks in advance.
0
votes
1answer
143 views

questions on bias of estimator

a) Let $X_{1},...,X_{n}$ be i.i.d Uniform$[0,\theta]$. Show that estimator $\beta(X)=max(X_{1},..,X_{n})$ is a biased estimator for $\theta$.Find an unbiased estimator, based on $\theta$. My attempt: ...
0
votes
1answer
40 views

a question which is somhow related to law of large number

suppose that $\mathbf p = [p_1, p_2, ..., p_n]'$ is a random vector. (' == transpose) and each element of $\mathbf p$ like $p_i$ is a Gaussian random variable with zero mean ($\mathbb E(p_i)=0$) and ...
0
votes
0answers
58 views

Extension to the Coupon Collector Problem

If there's n different coupons. Instead of ordering coupons one-by-one until you collect all n coupons as in the traditional 'Coupon Collector Problem', what if the coupons came in packs of m coupons. ...
1
vote
2answers
57 views

Prove variance in Uniform distribution (continuous)

I read in wikipedia article, variance is $\frac{1}{12}(b-a)^2$ , can anyone prove or show how can I derive this?