Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

0
votes
2answers
99 views

What is the trick in the derivation? Density of a complicated function

Through one of the proofs I found a problem that really cannot solve. Imagine some density function f(x). Now, imagine that the argument is a function of the form x+c(f'(x)/f(x)). Therefore, the ...
0
votes
1answer
777 views

Joint Uniform Distribution SOA question 93

This is a question from SOA: A family buys two policies from the same insurance company. Losses under the two policies are independent and have continuous uniform distributions on the interval from 0 ...
0
votes
1answer
816 views

Chi Squared Distribution with $\mu = 0$, $\sigma^2 \neq 1$

Let $X_i$ be independent normally distributed random variables with zero mean and variance $\sigma^2 \neq 1$. What is the probability density function of the random variable formed by the sum of their ...
0
votes
1answer
786 views

Distribution of the indicator function of a Gaussian variable.

Let $\xi$ be a random variable distributed according to a Normal distribution with given mean $\mu$ and standard deviation $\sigma$. Find the probability density function of $$ \psi = c\,\mathbb{1}_{...
0
votes
0answers
55 views

What is what between binomial many flip outcomes and statistical observation?

I am trying to get my head around normal distribution which evolves as a good approximation for a binomial problem (like coin flips). Theoretical outcome: When I have say 50 flips, there is a ...
0
votes
1answer
931 views

Let $T$ be exponential with parameter $\lambda$. Let $X$ be discrete defined by $X= k$ if $k \leq T < k+1$, $k=0,1,2,\dots$. Find the pdf of $X$.

To be honest, I am lost on this question. Here is what I have so far: $$ \ F_T(t)=- e^{-\lambda t}=P[T\le t] \ $$ $$ \ P[X=k]=P[k\le T \lt k+1] \ $$ I am not sure how to go about finding the pdf for $...
0
votes
1answer
36 views

Probability fail increase next attempt with variations

I have $3$ choices to make a attempt: first one costs $460$, and has a $10\%$ chance of success, and if it fails, it has a $3\%$ increase in chance for the next attempt. The second costs $1430$, and ...
0
votes
1answer
52 views

Calculate the densitiy of the random variable $X$

Given is $U \sim R(0,1)$ and $X=g(U) = -\frac{1}{\lambda} \ln(U), \space \space \lambda > 0$ What's the density of random variable $X$? $X \sim R(a,b)$ in general mean equal distribution. ...
0
votes
1answer
464 views

Conditional independence in Bayesian network with qualitative influences

I have some troubles solving an exercise from the book Probabilistic Graphical Models (pgm.stanford.edu). We are given the bayesian network with binary-valued variables. We do not know the CPDs, but ...
0
votes
1answer
601 views

pdf: What is the distribution of aX when X ~ Binomial / Gaussian

Question When $X$ is distributed as binomial or Gaussian, is $aX$ equivalent to some famous distribution? Here, $a$ is a real and positive number. Background I know a general formula giving $aX$'s ...
0
votes
1answer
112 views

How to obtain a pdf of a random variable defined as a function of many variables?

Given $N$ independent random variables ($X_1$,$X_2$,...,$X_N$) with individual pdfs $f_1$,...,$f_N$: How to determine the pdf of a random variable $Y=G(X_1,...,X_N)$?
0
votes
1answer
44 views

Equalitiy of two multinormal distributions

I have a problem proving the following thing: Consider a sequence of $r$ normal random variables $Z_j\sim N(0,1-p_j)$ (where $\sum_{i=1}^r p_i =1$) such that $E(Z_iZ_j)=-\sqrt{p_jp_i}$. On the hand ...
0
votes
1answer
1k views

How to scale a gamma distribution [closed]

I am trying to sample from a gamma distribution using transform sampling. All the example I seen as in the Wikipedia, is on the range from 0 to 20. How can I scale or extend the distribution over say ...
0
votes
3answers
320 views

How can two different random variables have the same probability mass functions?

I have a homework question that gives a function $PMF : [S → R] → [R → R]$ that maps random variables to their PMFs, and asks to show that it is not injective or surjective. With regards to the ...
0
votes
2answers
362 views

Probability that total weight of coffee in three 10-ounce jars is greater than the weight in one 30-ounce jar.

Suppose that instant coffee comes in two sizes, 10-ounce jars and 30-ounce jars. Let $X$ be the actual weight of coffee in a 10-ounce jar and assume that $X$ has a normal distribution with a mean 10....
0
votes
1answer
202 views

convergence to standard brownian motion

Could you help me with the following: I have that $$T(x):=\frac{X(nx)-E[X(nx)]}{\sqrt{n}} \xrightarrow{d} N(0, \frac{x^k}{k})$$ for each fixed $x>0$, where we also have that $\frac{X(nx)}{t}$ is ...
0
votes
1answer
521 views

Calculating the pdf and cdf of $X^2$ and $X^3$ with the pdf of $X$ given

Let $X$ be a random variable with the density funtion: $$\phi_X(x)= \begin{cases} 6x(1-x) & \text{if } 0<x<1, \\ 0 & \text{otherwise.} \end{cases}$$ Find the density and ...
0
votes
1answer
833 views

expectation of Gamma distribution help

If x∼Gamma(1,λ) how would i find the expected value E(e^bx) where b=aλ I'm kinda stuck as to how to approach the question. Some help will be greatly appreciated Thank you in advance
0
votes
1answer
1k views

What is the closed-form expression for “cumulative density of a zero-mean unit-variance Gaussian” and one other?

I'm working on a hobby project that involves tournament-style player rankings, and I'm using the TrueSkill system developed by Microsoft for online gaming. It's a great system, but much of the math ...
0
votes
1answer
79 views

Find pdf of sum of n indp exp RVs w/o using MGFs

From Williams' Probability w/ Martingales: Re $E[f(S_n)]$, how do I obtain $f_{S_n}(s)$? It seems that $$f_{S_n}(s) = \frac{s^{n-1} e^{-\lambda s} \lambda^n}{(n-1)!}$$ I tried computing $M_{S_n}(t) ...
0
votes
0answers
168 views

How to construct “destandardized” t distribution?

Suppose I have a sampling distribution, with sufficiently large sample size, that due to CLT, I have good guassian approximation as below with 95% CI. The actual distribution that forms after many ...
0
votes
1answer
480 views

Moment generating function (MGF) of the ratio distribution $\displaystyle\frac{X}{Y}$

If we know the moment generating functions (MGFs) of the random variables $X$ and $Y$ to be $M_{X}(s)$ and $M_{Y}(s)$, respectively. The MGF of the sum $X+Y$ will $M_{X}(s) \cdot M_{Y}(s)$. So what ...
0
votes
2answers
56 views

Trouble deriving sum of squared normals is Exponential with mean $2$

Box-Muller method hinges on the fact that $R = Z_1^2 + Z_2^2$ is Exponential with mean 2, where $Z_1, Z_2$ are independent standard normals. I want to derive this fact but am getting stuck. I proceed ...
-1
votes
2answers
589 views

How do I solve the questions below using the Poisson distribution with the variable $t$? [closed]

This is the problem I have to solve for a job at school. Can anyone help me, what kind of distributions approximations do I use and : In (a), how do I manipulate t in order to find the desired ...
-1
votes
1answer
344 views

Proabability Bayes theorem cancer question

Cancer is present in 22% of a population and is not present in the remaining 78%. An imperfect clinical test successfully detects the disease and with probability 0.70. Thus, if a person has the ...
-1
votes
2answers
2k views

Calculating moment generating function with normal distribution [closed]

My problem is to calculate $E(e^{\lambda X})$, where X has normal distribution $N(\mu, \sigma^2)$. So I have to calculate integral $\int_{-\infty}^\infty e^{\lambda x} \frac{1}{\sqrt{2\sigma^2 \pi}} ...
-1
votes
2answers
82 views

Determine the joint distribution of a random vector in $\Re^2$

A random vector in $\Re^2$ is chosen as folllows: Its length, $Z$, and its angle, $\Theta$, with the positive $x$-axis, are independent random variables. $Z$ has density $$f(z)=ze^{-z^2/2}, z>0$$ ...
-1
votes
2answers
190 views

Integral of an integral with variable limits

I'd like to prove the following but not sure where to start: $$\int_{-\infty}^\infty\int_{-\infty}^a\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{x^2}{2}\right)dx\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{...
-1
votes
1answer
70 views

How to model this distribution using Central Limit theorem?

I have distribution that can be defined as below, $S=a_0\cdot b_0 + a_1\cdot b_1 + a_2\cdot b_2 + \cdots +a_{n-1}\cdot b_{n-1}$ Now, I want find the distribution of $S$ when, $a_i$'s are selected ...
-1
votes
2answers
2k views

Related To Binomial Distribution

A blackjack player at a Las Vegas casino learned that the house will provide a free room if play is for four hours at an average bet of $50. The player’s strategy provides a probability of .49 ...
-1
votes
1answer
308 views

Finding distribution function of the ratio of two continuous uniform random variables where the denominator random variable is squared.

Let $X_{1}$ and $X_{2}$ be independent and uniformly distributed between 0 and 1. I want to find the distribution function of $X_{3}=\dfrac{X_{2}}{X_{1}^{2}}$. Denote this distibution function by $...
-2
votes
1answer
33 views

PMF from two discrete probabilities [closed]

If I have two probabilities, $P(X>n)$, and $P(X>n+m)$, how would I go about obtaining a PMF of some random variable $X$? This problem seems simple enough but I am totally lost on how to arrive ...
-2
votes
1answer
2k views

What is the smallest number of socks you should pull out so that you can be assured that you will have at least one pair of matching socks? [closed]

Anyone can help me with this math problem? Your drawer has 5 pairs of black socks, 4 pairs of gray socks, 2 pairs of white socks, 1 pair of brown socks, and 1 pair of blue socks. The lights are ...
-5
votes
1answer
152 views

Question about Markov moment and $\sigma$-algebra [closed]

Let $\tau$ be the Markov moment with respect to the stream $(\mathcal{F}_{t}, t \in T)$. Prove that $$ \mathcal{F}_{\tau}=\{A \in \mathcal{F}: A \cap \{ \tau \leq t \} \in \mathcal{F}_t, \quad \...
-6
votes
2answers
283 views

Comparing uniform random variables.

$X$ is a uniformly distributed random variable on $(0,1)$ $Y$ is a uniformly distributed random variable on $(0,2)$ $Z$ is a uniformly distributed random variable on $(0,4)$ What is the probability ...