2
votes
1answer
100 views

How do I find the PMF of X when X is the number of flips of a fair coin that are required to observe the same face on consecutive flips?

How do I find the PMF of $X$ when $X$ equals number of flips of a fair coin that are required to observe the same face on consecutive flips? The hint was to draw some sort of a tree diagram, but I'm ...
0
votes
0answers
10 views

A cell dies at a constant rate r and what is the density function of its life time.

The problem: A cell dies at a constant rate $r$ and the its life time is the duration from t=0 to when when it dies. what is the density function of its life time $l$? I have done some relevant ...
3
votes
2answers
66 views

In composition of two mappings, can the outer mapping access the arguments of the inner mapping?

In composition of two mappings, can the outer mapping access the arguments of the inner mapping? Here is an example to illustrate my question and my thought. E.g. $f: \cup_{n \in \mathbb N} \mathbb ...
1
vote
1answer
33 views

Can this function be a density function of a continuous random variable X?

F(x) = 0, if x < 1 F(x) = 1, if 1<=x<=2 F(x) = 0, if x>2 I think it could be, as long as the integral is 1. Any ideas?
1
vote
1answer
62 views

Statistics Probability Density Functions with Mutliple Features (Multivariate Normal Distribution)

I'm looking for a beginner-friendly explanation on how this Probability Density function works when dealing with mutliple features and what the variables and terms mean in detail. I'm seriously ...
0
votes
1answer
60 views

Integrability condition

Suppose that \begin{align} \mathbb{E}\int_{0}^{T}f^{2}(t)dt <K \end{align} Does it also hold that \begin{align} \int_{0}^{T}f^{2}(t)dt <K \end{align} ? Here, T, K>0 are fixed. I am a bit ...
1
vote
1answer
25 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
vote
3answers
223 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
34 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
59 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
58 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
vote
0answers
22 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
65 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
45 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
45 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
57 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
262 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
147 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
13 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
35 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
148 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
53 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
23 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
26 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
95 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
35 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
28 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, ...
2
votes
0answers
53 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
50 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
30 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
154 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
119 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
106 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
54 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
120 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
195 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
33 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
79 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
40 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
28 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
53 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
28 views
2
votes
1answer
87 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
60 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
59 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
113 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: ...