Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

learn more… | top users | synonyms

3
votes
1answer
26 views

Is the chance of a variable also a parameter for a probability distribution?

I'm new to statistics and I'm a bit confused about the concepts of 'chance of a variable' and 'parameters of a probability distribition'. Is chance also a parameter? And if so: can computing the ...
0
votes
0answers
46 views

Breaking a stick randomly at two points: Expected value of the largest piece. [duplicate]

If a stick of unit length is broken randomly at two points, to make 3 pieces of stick. What is the expected value of the largest stick. Is there an elegant solution to this problem? Thanks.
3
votes
1answer
59 views

$\sum X_n$ converges a.s.

This one is from old qualifying exam. $\{X_n\}$ be non-negative, independent and $\{Y_n\}$ is another sequence (not necessarily independent) but $X_n \sim^{d} Y_n$. Then $\sum X_n$ converges a.s. ...
1
vote
1answer
24 views

“Degrees of freedom” of a variable with Chi-Square distribution

The gamma distribution with parameters $\alpha$ and $\beta$ is: $$f(x) = \frac{1}{\beta^{\alpha} \Gamma(\alpha)} x^{\alpha - 1} e^{-x/\beta}$$ When we substitute $\alpha = r/2$ and $\beta = 2$, we ...
1
vote
2answers
71 views

Proving that for any Differentiable distribution $F(x)$, an expression is increasing in $x$?

I am guessing that for a continuous random variable on $[0,1]$, $$ U(x)=\Big[x F(x) + \int_x^1 (1-t)f(t)dt\Big]x $$ is increasing for any distributions, because I can show $$ U'(x)=2xF+x^2f+\int_x^1 ...
1
vote
0answers
50 views

Integrating a prob distr over the set of possible circles within an annulus

Let $z$ be the measured coordinates of a point on a circle $c$ with center $x$ and radius $r$. Assume the probability of measuring $z$ given the circle $c$ is normally distributed by the distance ...
0
votes
1answer
25 views

I want to find PDF by differentiating CDF and then from PDF, expected values of the following problem. .

Three light bulbs have independent exponentially distributed lifetimes with a common parameter $\lambda$. What is the probability distributed function and expected value of the time until the last ...
0
votes
1answer
40 views

Probability for the minimal polynomial to be equal to the characteristic polynomial

Consider the space $M_n(\mathbb R)$ of real square matrices of dimension $n$. Is there a way to define the probability for the minimal polynomial to be equal to the characteristic polynomial? By a ...
2
votes
2answers
35 views

a problem regarding conditional probability and binomial distribution.

Die A has 4 red and 2 white faces whereas die B has 2 red and 4 white faces . A coin (fair) is tossed once . If it falls head , the game is carried on by throwing die A alone. If it falls tail die B ...
1
vote
2answers
40 views

Exponential Distribution ( Probability Problem ).

We know that probability density function $f(x)$ for an exponential distribution with parameter $\lambda$ is given by : $f(x)= \lambda e^{- \lambda x}$ We are given the following question : If the ...
1
vote
2answers
27 views

What's $r$ going to be when you get the summation of $36$ Geometric $X_i$'s

Let $X_1,X_2,\ldots,X_{36}$ be a random sample of size $n=36$ from the geometric distribution with the p.d.f: $$f(x) = \left(\frac{1}{4}\right)^{x-1} \left(\frac{3}{4}\right), x = 0,1,2,\ldots$$ Now ...
1
vote
1answer
25 views

Find the distribution law of a function of a random variable

Let $X$ be a random variable with an exponential distribution $X\sim\operatorname{Exp}(\lambda)$, such that its expected value $\mathbb E[X] = 2$. Let $f$ be a function such that: $$f(x) ...
1
vote
0answers
14 views

Prime Factorisation Probability Decrease: upper bound

Suppose we have a probability distribution $p$ over $\{0, 1, 2\}$, with probabilities $p_0$, $p_1$ and $p_2$, and $p_0 + p_1 + p_2 = 1$. Now suppose we repeatedly choose an element randomly from this ...
0
votes
1answer
43 views

probability distribution for each step in a drunkards walk

Imagine a typical drunkards walk (2D) made of steps $\ell$ each of length $L$ in any direction. I was told that the probability distribution of each step can be written as a Dirac delta like this ...
0
votes
1answer
25 views

Coupon Collector with different probabilities

I recently came across the following problem: Bonnie wants to collect 2 coupons. Each box costs 10\$, and she doesn't know which coupon will be inside before she opens it. The probability of ...
-1
votes
0answers
48 views

In a generalized birthday problem, how is this simplification done?

I am trying to understand generalization of birthday paradox in probability as it is explained here. I think I got the whole solution except the below simplification. ...
0
votes
0answers
14 views

What is the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A company has a one-year group life policy that divides its employees into two classes as follows: Class, Probability of Death, Benefit, Number in Class, A, 0.01, ...
0
votes
2answers
21 views

Is XY the joint distribution over X and Y?

The definition of covariance is $Cov(X,Y) = E[XY] - E[X]E[Y]$ I can't wrap my head around what $XY$ is supposed to be. I suspected it to be the joint distribution over $X$ and $Y$ but I could not ...
-2
votes
1answer
79 views

Very hard probability problem [closed]

One of the first 6 positive integers is to be chosen by casting an unbiased die. Let this random experiment be repeated five independent times. Let $X_1$ be the random variable representing the number ...
0
votes
0answers
15 views

distribution of infinite sum of independent but non-identical normal variables

For $i=1,2,\ldots,n$, suppose $X_i \sim N(0,\Omega_{i})$, where $\Omega_{i}$ is of dimension $k\times k$. It is known that $\frac{1}{\sqrt{n}} \sum_{i=1}^{n} X_i \sim N(0, \overline{\Omega})$, where ...
3
votes
0answers
99 views
0
votes
0answers
14 views

what do we get if we compare fisher information of two distributions?

What do we get if we compare fisher information of two distributions belonging to different family of distributions with shape and scale parameters say we compare it for generalised exponential and ...
1
vote
3answers
51 views

Finding maximum of two variables

Given $X$ is uniform on $[0, 10]$. Let $$Y = \max(5, X).$$ Determine Var(Y). I'm familiar with how to find the variance of a uniform random variable, as well as the max of two random variables. ...
0
votes
0answers
10 views

$\exists \mathcal{A},\mathcal{B}:X\sim \mathcal{A}\Rightarrow \frac{p}{\sqrt{q+rX}}\sim \mathcal{B}$?

Does there exist a parametric distribution $\mathcal{A}$, such that: $X\sim \mathcal{A}\Rightarrow\frac{p}{\sqrt{q+rX}}\sim \mathcal{B}$ for some parametric distribution $\mathcal{B}$ Where ...
-1
votes
0answers
31 views

what is distance between two random point in two non overlapping circles?

I have a large circle with radius R that two set poisson random points with different densities in inside circle distributed. there are small circles to radius Rc in this large circle that points ...
1
vote
1answer
21 views

A die is cast $3$ independent times, let $Y = \max(X_1,X_2,X_3,)$…

A die is cast $3$ independent times. Let $X_i$ be the random variable representing the number on the face appearing at the $i$th cast. Let $Y$ be the random variable defined by $Y = ...
0
votes
2answers
24 views

Finding standard deviation given joint probabilities

I'm trying to find the standard deviation of $Z = X + Y$ given the following table: I'm getting $E[Z^2] = 36.31$ and $E[Z] = 5.45$, giving me a variance of $Var[X] = 36.31 - (5.45)^2 = 6.6075$ ...
3
votes
0answers
39 views

PDF of $X = \max\{X_1,X_2\}$, being $X_1$ and $X_2$ independent Normal distributed random variables

Let $X_1 \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $X_2 \sim \mathcal{N}(\mu_2,\sigma_2^2)$, what's the CDF of $X = \max\{X_1,X_2\}$? Both variables are assumed to be independent. I tried the ...
0
votes
1answer
25 views

How does sampling affect the distribution of frequencies of individual types?

Consider a population of size N in which individuals can be of x different types. Take a sample (with replacement) of size ...
2
votes
1answer
28 views

Probability of achieving Maximum Value in a Probability Density Function

As part of my (incomplete) lecture notes I've been provided with this example but, alas, with no supplemented solution. As this is the first (& only) example following on from Theorem 1.4 ...
2
votes
2answers
54 views

If X is log-normal, is: $\frac{a}{\sqrt{b+cX}}$?

I am working for the first time with log-normal distributions and I want to verify whether the following statement is true. I am not sure whether all the properties of the log-normal distribution hold ...
3
votes
1answer
54 views

Optimal algorithm for guessing random variable

Let's say you have some unknown quantity $$X\in [0,1]$$ We have N tries to guess the value of X - if you guess $$g_{i}\le X$$ then you capture value $$V_{i} = g_{i}$$ while if your guess is over the ...
-1
votes
1answer
37 views

random distribution/probability

Let $a$ and $b$ be positive integers with $a \le b$, and let $X$ be a random variable that takes as values, with equal probability, the powers of $2$ in the interval $[2^a, 2^b]$. The question is ...
0
votes
0answers
16 views

probabilistic rbotics

we will apply Bayes rule to Gaussians. Suppose we are a mobile robot who lives on a long straight road. Our location x will simply be the position along this road. Now suppose that initially, we ...
0
votes
1answer
28 views

Deduce probability density function from integrals on $N$ bounded regions

I stumbled upon this problem. Suppose you have an unknown probability density function on the two dimensional real plane, $f(x,y)$ Suppose you have three distinct points $P_1, P_2, P_3$ on the plane ...
0
votes
0answers
19 views

Minimax of Negative Binomial

Studying for the quals and I am in a deep confusion. This is an exercise from Mathematical Statistics: A Decision Theoretic Approach by Thomas S. Ferguson. Let $X$ has density $$f(x|\theta)={r+x-1 ...
1
vote
1answer
53 views

What does statistical significance literally mean? [closed]

I am getting confused when trying to determine the statistical significance of my results. For instance, I was attempting to make a conclusion of reliability of a model after 100000 trials depending ...
1
vote
2answers
26 views

Does logarithm of Gaussian image still gaussian distribution?

I have an image 2D that pixel intensity follows multi Gaussian distribution such as $$p \left( I(x) \in \Omega_i \mid (I(x)\right)=\frac{1}{2\pi \sigma_i}\exp\left(-\frac ...
0
votes
1answer
27 views

Deriving Mean for Negative Binomial Distribution.

A negative binomial distribution is given by : $P(X=x)\:: \binom{x-1}{k-1}p^{k}(1-p)^{(x-k)}$ , where p is probability of a success. where , x = k , k+1 , k+2 ,..... and so on. Mean is given by : ...
0
votes
1answer
19 views

Find $P(796.2 \leq \sum_{i=1}^{16}(X_i-50)^2 \leq 2630)$ with a sample of $n=16$ and $X \sim N(50,100)$

If $X_1,X_2, ..., X_{16}$ is a random sample of size $n=16$ from the Normal Distribution $N(50,100)$, determine: $$P(796.2 \leq \sum_{i=1}^{16}(X_i-50)^2 \leq 2630)$$ Okay well I know that ...
0
votes
3answers
57 views

Probability of a Poisson random variable taking a specific value [closed]

Let $X$ be a random variable with Poisson distribution with mean $\lambda=9$. Knowing that $P(X=8)=0.131756$, compute $P(X=9)$. How do I approach this problem?
1
vote
2answers
37 views

Uniqueness of moments for probability distributions with infinite moments.

I was taught the collection of a distribution's moments uniquely defined the distribution. Recently, I have been studying Pareto distributions, which have infinite means for shape parameters less than ...
-1
votes
1answer
53 views

Problem with Binomial distribution

Can you help me with this problem - how to solve it or what methods to use? The amount of burnt lamps in a device is a binomially distributed random variable $$X \sim \mathcal{Bin}(3, 1/3)$$ The ...
0
votes
1answer
36 views

Probability that 5 out of 7 bulbs will produce white flowers

If you have a bag with 25 tulip bulbs that will grow into white, yellow or red flowers. You want to plant 7 bulbs. Each bulb in the bag, independently of the others, grows into a white tulip ...
2
votes
1answer
29 views

stochastic dominance definition

I was wondering if, for positive random variables $X$ and $Y$, $\Pr(X\geq Y)\geq 1/2$ implies $\Pr(X\geq x)\geq \Pr(Y\geq x)$. Intuitively it "makes sense", since $X$ tends to be more often bigger ...
1
vote
1answer
33 views

$\frac{\chi^2_n}{n}$ Stochastically increasing in $n$?

I was wondering whether $\frac{\chi^2_n}{n}$ is stochastically increasing in $n$. My main problem: Suppose $\hspace{5pt}\frac{(n-p)\hat{\sigma}^2}{\sigma^2} \sim \chi^2_{n-p}$. Then the expected ...
1
vote
1answer
32 views

Proving that the variance is non-negative

$$(E(X))^2 = \left( \int_{-\infty}^{+\infty}xf(x)dx \right)^2 \le \int_{-\infty}^{+\infty}x^2(f(x))^2dx \le \int_{-\infty}^{+\infty}x^2f(x)dx = E(X^2)$$ Because of cauchy-schwarz inequality and $f(x) ...
0
votes
1answer
25 views

Given the moment generating function of a continuous-type r.v, how to find the p.d.f?

Say for $t<1$: $$M(t) = \frac{1}{(1-t)^2}$$ How to find the p.d.f of the random variable? $$M(t) = E(e^{tx})=\int_{-\infty}^{+\infty}e^{tx}f(x)dx$$ How do we find: $f(x) = xe^{-x}$ on ...
2
votes
3answers
43 views

Distribution of sum of random variables

Let $X_1, X_2, . . .$ be independent exponential random variables with mean $1/\mu$ and let $N$ be a discrete random variable with $P(N = k) = (1 − p)p^{k-1}$ for $k = 1, 2, . . . $ where $0 ≤ p < ...
0
votes
2answers
38 views

Neighbor Interaction in a Random List

Assuming a random arbitrarily long list where each element has a $50\%$ chance of being a $0$ or a $1$, such as: $0001101101$ What is the chance of having a neighbor that isn't the same? For ...