1
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1answer
15 views

Convex and Concave Functions using Known Function Values

I am reading the classic Prospect Theory: An Analysis of Decision Under Risk (1979, Econometrica) by Kahneman and Tversky. I am not clear on something on page 278: ...
1
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3answers
202 views

Expressing the probability density function of $Ax$ in terms of the pdf of $x$

I understand that, for example, you might have a density function which measures the probability of observing an outcome in a certain interval measured in feet, but someone wishes to use meters ...
0
votes
1answer
15 views

Intersection of 2 Indicator Functions

Let $E$ and $F$ be events. Let $I_E(\omega)= \left\{\begin{array}{cc} 1, & \omega\in E, \\ 0, &\omega\in E^C. \end{array}\right.$ Show that $I_{E\cap F}(\omega)=I_EI_F$ I found the answer ...
1
vote
0answers
49 views

Conditioning on function of random variable and random variable itself

Suppose that $Y_{i}\in\{0,1\}$ is a binary variable, and $X_{i}$ is some random vector in $\mathbb{R}^{d}$ . Why can we say the following: \begin{eqnarray*} ...
1
vote
2answers
55 views

limit of $ f(n) = 100 \left (1 - \frac{1}{n}\right) ^{ n}$

So I was daydreaming about math (like I do frequently) and I came up with this question/riddle. Say you have a die. If you roll a 1 you lose, otherwise, you win. This die has n sides on it, and you ...
1
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0answers
18 views

Tips for finding the range of a function

I am studying probability, specifically joint probability distributions. When computing sums or quotients I end up with things like this (when working with uniform random variables for instance): ...
2
votes
0answers
60 views

Using the ELO Rating System on Static Objects

The Setup Suppose we have a list of movies $m_1, m_2, \dots, m_n$ that we wish to rank in order of "quality." We define the "strength" of a movie $a$ by a function $f$ which takes in numerical ...
1
vote
1answer
37 views

What are the chances of getting at least 3 fours when rolling 5 dice?

What are the chances of getting at least three or more "fours" or higher number when rolling a fair six-sided die five times? Actually, this is a problem I am curious about the answer and I can't ...
1
vote
0answers
44 views

Function for amplification of probabilities

To simplify my task, let's just say that we have a situation when we have to choose a point and we have N=2:7 points to choose from. We have a criteria by which ...
0
votes
3answers
50 views

2 independent poisson random variables probabilities and 2 different proofs

So, in the above exercise I was wondering if I could get some help with : 2.1 - I was told moment generating functions could help me prove that, but I can't get it 2.2 - I don't get how to start ...
2
votes
2answers
258 views

what is the probability that the selected function maps prime numbers to prime numbers?

Let $X = {1, 2, 3, . . . , 25}$. If a student selects a function randomly from the set of all functions from X onto X, then what is the probability that the selected function maps prime numbers to ...
0
votes
3answers
72 views

How to put probability density function in C++?

I have a random variable X that has a probability density function of f(x)=x^(-1/2)/2 for all x between 0 and 1. double RANDOM; I need to give a value to RANDOM, that accepts the PDF All that I ...
1
vote
0answers
12 views

unimodality and continuous

i would like to ask question about unimodality of probability function ,from wikipedia http://en.wikipedia.org/wiki/Unimodal it says that In mathematics, unimodality means possessing a unique mode. ...
1
vote
1answer
46 views

Uniform distribtion: clarification of $f_X(x)$

I have $Y=2(X-1)^2 -1$ where $X$ is uniform distributed on $[0,2]$ I want to find the pdf of $Y$ and expected value of $Y$. My question is just: Does $X$ have pdf $f_X(x)= \frac{1}{2}$?
0
votes
1answer
34 views

Cumulative distribution function picture problem

this is from an past exam paper I got for part 1: O≤X≤r = x^2*π /r^2*π and how do I do part 2 is that right, also I don't really understand the question, when it talks about circle in the question, ...
1
vote
2answers
133 views

How to find binomial pmf with probability = another random variable

Assume that $Q$ is a random variable with density proportional to $q$ for $0 < q < 1$. Given $Q = q$, $N$ has a binomial distribution with parameters $n$ and $q$. What is the probability mass ...
2
votes
1answer
42 views

Using a Moment Generating Function to find a probability function

I'm struggling hugely with breaking down my M.G.F. into something that I can use to give me a probability function of $X$, the problem reads: Find the probability function, $f$, of $X$ including ...
0
votes
1answer
19 views

Finding the probability density function for IID rv [duplicate]

The question is as follows: Suppose that X1 and X2 are independent, identically distributed exponential random variables. Determine the PDF for for X1 - X2. I understand that because X1 and X2 are ...
0
votes
0answers
25 views

Is it possible to show that the given equation is monotone?

I want to know if $$\int_{l_l<l<l_u}\left(l_l^{-1}l\right)^{\Large\frac{\ln\left(k(l_l,l_u)\right)}{\ln\left(l_u/l_l\right)}}f_1\mbox{d}\mu$$ is monotone in $l_l$ and/or $l_u$. Given: ...
2
votes
1answer
63 views

Working out the median of a beta function

I am trying to work out the median of the beta function of $\mathrm{B}(1/2,1/6)$. I have been told the answer to this is $0.9510$ but i'm unsure to get there? Is there a simple formula in order to get ...
0
votes
1answer
21 views

How many functions defined on $n$ points are possible if each functional value is either $0$ or $1$?

How many functions defined on $n$ points are possible if each functional value is either $0$ or $1$? This is from the text A First Course on Probability by Sheldon Ross. The solution he ...
0
votes
0answers
27 views

Function plotting

I have a function $f(x)=\binom{N}{K} \ln(1-F(x)), x \geq 0$, where $F(x)$ is a cumulative distribution function. Then, $\ln(1-F(x))$ is negative for various values of $x$ as $F(x) \geq 0$. Also, ...
1
vote
0answers
45 views

Is this function monotonically non-decreasing?

I am wondering if the function $L[n]$ defined on $n=0,1,2,\ldots,N$ below is "monotonically" non-decreasing in $n$. I put monotonically in quotes because the function is not continuous and I am not ...
0
votes
1answer
48 views

Specify a function that majorizes $\frac{2}{\pi}\sqrt{1-x^2}, -1\leq x \leq 1$ [closed]

Specify a function that majorizes $\frac{2}{\pi}\sqrt{1-x^2}, -1\leq x \leq 1$ Could anyone please help me? I dont have a clue how to start.
0
votes
1answer
28 views

Find pdf of $f(x)$ such that $g(x)/f(x)$ is approximately a constant

My friend asked me a question that asks to find a pdf function $f(x)$ such that $f(x)/g(x)$ is approximately a constant, where $g(x)=\sqrt{e^{x^2}+e^x}$, and $f(x) \neq g(x)$. And the range of x is ...
0
votes
0answers
121 views

Solving sample size of hypergeometric distribution given a specific probability

I am trying to figure out how to calculate the sample size of a hypergeometric distribution, given a population, population successes, and probability. Here is the initial formula: ...
1
vote
0answers
67 views

Probability Density Function problem

$$ f(x) = \begin{cases} 0 & \text{if $x < 0$} \\ x^2 & \text{if $0 ≤ x < \mu$} \\ α + βx & \text{if $\mu ≤ x ≤ 10$} \\ 0 & \text{if $x ≥ 10$} \end{cases} $$ Considering ...
3
votes
1answer
105 views

$n^n$ are the moments of a measure on the non-negative real line?

I would like to know if the numbers $1,1,2^2,3^3,\dots, n^n,\dots$ are the moments with respect some measure $\mu$ on $[0,+\infty)$, i.e., if there exists such a measure $\mu$ with $$n^n=\int_0^\infty ...
3
votes
1answer
52 views

probability of a function f(x) to be increasing

Suppose $f(x)=x^3+ ax^2 + bx +c$ . Now a,b,c are chosen respectively by throwing a dice 3 times. Now find the Probability that $f(x)$ is a increasing function ? MY APPROACH : i really have given a ...
0
votes
1answer
78 views

Generating Random Serialnumber with least similarity

I want to generate 16-digits hexadecimal serial-number like: F204-8BE2-17A2-CFF3. (This pattern give me 16^16 distinct serial-number But I don't need all of them as I describe below) I need an idea ...
3
votes
1answer
124 views

Exponentially bounded

What do you mean by a function being exponentially bounded? In the context of fat tailed distributions; tail being not exponentially bounded.
3
votes
4answers
81 views

Finding generating functions - how was this jump made?

I'm going through examples of probability-generating functions in a book and am confused by the following example: $$1+2s+4s^2+...=\sum_{n=0}^\infty (2s)^n=(1-2s)^{-1}$$ I understand the summation but ...
0
votes
1answer
32 views

Function of a continuous random variable

Random variable X have a density of distribution Density of a random variable $Y = 1-X^2$ I started solving this, but I'm not sure is my solution correct and if not where is my mistake? $$ ...
0
votes
2answers
64 views

The bounds for a Joint Probability Function

For $$ f(x,y) = \begin{cases}\frac{1}{y} & 0 < x < y < 1\\ 0 & \text{elsewhere} \end{cases} $$ find $P(X + Y > 1/2)$ First i should make it $1 - ...
1
vote
0answers
38 views

A marginal density function problem

Given a plane with three points, $(0,−1), (2,0)$, and $(0,1)$ with $x$-axis and $y$-axis connecting three points to make a triangle. Suppose this triangle represents the support for a joint continuous ...
1
vote
2answers
26 views

Expectation of function of two random variables

I have the random variables X and Y, with joint density function $f(x,y)$ over the plane $-\infty < x < \infty$ and $-\infty < y < \infty$. I am trying to find the expectation of ...
1
vote
0answers
19 views

How can I find the probability of a value occuring in an interval of a given step function?

Knowing a step function $N(t)$ defined over the interval $[0,T]$, which takes values in $0, 1, ..., k$, how can I define the probability $p(n)$ for $0 \leq n \leq k$, which is the probability of a ...
1
vote
2answers
50 views

How to calculate $E[X^2Y^5]$ given density functions for $x$ and $y$

Let $X$ and $Y$ be random independent variables within the limits $[0, 1]$ with the following density functions: $f_X(x) = 0.16x + 0.92$ such that $x$ is within the parameters $[0, 1]$ and $f_Y(y) = ...
1
vote
1answer
27 views
2
votes
1answer
84 views

Difficult Discrete/Probability Problem

Here's the question: For a function $f:[n]\rightarrow[n]$, where $n$ is the set $\{1,2,3,\ldots,n\}$, define the inverse complexity, $ic(f)$ as the number of ordered pairs $\langle i,j \rangle$ ...
2
votes
1answer
45 views

Variance of the Random Variable $|im(f)|$ where $f:[n] \rightarrow [n]$

Here's a question: Let $f$ be picked randomly from the set of all functions from $[n]$ to $[n]$, where $[n]$ is the set $\{1,2,3,\ldots,n\}$. Give a closed-form expression for the variance of the ...
0
votes
1answer
57 views

Autocorrelation functions of 2 correlated stationairy processes

I have some trouble solving the following problem: Given are the stationairy processes $X_t$ and $Y_t$: $X_t = Z_t*\sqrt{7+0.5X_{t-1}^2}$ $Y_t = 2+(2/3)*Y_{t-1}+X_t$ Where $Z_t$ is distributed IID ...
0
votes
1answer
58 views

Combinations and probability: probability density function

The probability density function of $X$, the lifetime of a certain type of electronic device (measured in hours), is given by $f(x)=xe^{-x}$ for $x\in[0,\infty)$. Find $P(X>2)$. Please ...
1
vote
3answers
108 views

Given the joint distribution of two random variables, compute the probability that one is less than the other?

Let $X$, $Y$ have the joint density function $$f(x,y) = \frac{1}{2\pi} e^{-(x^2+y^2)/2}$$ Compute $P(X<Y)$. I believe that I should set up a double integral over this function, like so: ...
0
votes
2answers
76 views

PMF : Determine the distribution function of X

The spectrum of a discrete random variable X consists of the points 1, 2, 3,..., n and its probability mass function (pmf) fi = P(X = i) is proportional to 1/i(i+1). Determine the distribution ...
0
votes
1answer
34 views

Probability and time

I've heard somewhere from someone about a theorem that roughly says "the probability of an event decreases as time increases" I couldn't find the exact theorem (assign it exists at all.) So figure I ...
0
votes
2answers
709 views

cdf/pmf/pdf validity question

Studying for a statistics exam. I have come across this problem: and it presents to me some important and extremely basic questions (I have a LONG way to go before I'm prepared for this exam). ...
3
votes
1answer
363 views

Calculations for a random variable given density function

Studying for a statistics course an stumped on how to go bout solving a problem. The following is a problem I have solved - the problem I'm having trouble with is related to this one: ...
1
vote
1answer
94 views

Ratio of convex functions with dominating derivatives is convex?

Let $f,g:\mathbb [0,\infty)\rightarrow (0,\infty)$ satisfy $f^{(n)}(x)\geq g^{(n)}(x)>0$ for all $n=0,1,2,\ldots$ and $x\in [0,\infty)$. In particular, $f\geq g> 0$ are increasing and convex ...
1
vote
3answers
82 views

Self multiplication of a CDF degenerates into a Dirac Delta?

Because of some work I need to do, I bumped into this problem. A CDF $F(x):\mathbb{R} \mapsto [0,1]$ of some random variable is raised to the n-th power. The function is continuous all over its domain ...