# 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.

17,107 questions
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### Does MLE really care about PDF?

I am wondering, whether MLE really cares whether it operates on proper distributions. Lets take a look at the following situation: likelihood: $$L(\theta \mid x) = \prod_{n}^{N}{f(x_n \mid \theta)}$$ ...
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### Random Variable expression with the distribution function

In my textbook, it is said that if we chose the probability space ( (0,1) , B(0,1) , P) where P is the Lebesgue measure, and we have the distribution function FX(x) of a random variable X we can find ...
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### MLE of simultaneous exponential distributions

Given the $X_i\sim \text{exp}({\theta})$ and $Y_i\sim \text{exp}(\frac{1}{\theta})$, where $X_i$ and $Y_i$ are indpendent, with the same $\theta>0$. I have to find the MLE and its distribution. I ...
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### Existence of a sequence of independent $E$-valued random variables with distribution $\mu$ given $\mu$ and $E$ Polish

I know that the following question is true for $E=\mathbb{R}$. I would like to know if it can be extended to Polish spaces. Suppose that $(E,d)$ is a Polish space. Write $\mathcal{B}(E)$ for the ...
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### Independence of sample mean and sample variance

It is well known that under normality assumption, the sample mean and sample variance are independent, by Basu's Theorem. My question is that, is the normal distribution the only distribution whose ...
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### Sampling subsets of positive numbers with sum close to a given number

Given a set of $N$ positive numbers $a_1 \ldots a_N$, I am trying to generate random subsets of this set such that the sum of numbers in subset is close to a given number $M$. Suppose I can only ...
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### Expectation of sum of geometric random variables vs. expectation of Pascal r.v.

Let $\{X_i\}$ be a Bernoulli process, i.e. $X_1, X_2, X_3, \dots$ are i.i.d. Bernoulli variables with parameter $p$. Let $T_k$ be the time at which the $k$th success occurs. I can reason about the ...
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### Convergence of Random Variable (convergence in probability)

Let $(x_{n} )$be a sequence of real random variable defined on probability space ,converge in Probability to $x$ . Let $y$ be a random real variable on probability space. For $\varepsilon>0$ ...
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### Convergence of random variable ! (Probability) [on hold]

Let $(x_{n} )$be a sequence of real random variable defined on probability space ,converge in Probability to $x$ . Let $y$ be a random real variable on probability space. For $\varepsilon>0$ ...
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### Inference regarding the mean lifetime of a bulb using a new technique

The lifetime in hours of each bulb manufactured by a particular company follows an independent exponential distribution with mean $\lambda$. We need to test the null hypothesis $H_0: \lambda=1000$ ...
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### Probability generating function of exponential distribution

The exponential distribution is given by: $$PDF: \lambda e^{\lambda x}$$ And the formula for probability generating function is given by: $$G(z) = \sum_{x=0}^\infty p(x)z^x$$ where $p(x)$ is a ...
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### Lexicographical order in context of identifiability of mixture of two Normal distributions

I want to understand a method used in a paper on identifiability of mixture of two Normal distributions. This is Teicher 1963 "Identifiability of finite mixtures", fragment of the proof The author ...
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### How to determine Y(n)

A random variable $x$ from the set $\{1, 2, ... ,n\}.$ Let $x$ has distribution function $f(k) = Y(n) · g^k$ where $g$ is a fixed number within $0$ and $1$. Find $Y(n)$ which is a constant term in ...
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### Problem to calculate a marginal function in probability

I have a problem in probability. I have $f(x, y) = \frac 14 \cos(y)$, if $x$ is between $0$ and $\pi$, and if$y$ is between $-\frac x2$ and $\frac x2$. I have to calculate $f(y)$. I calculated ...
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### Linear transform of bivariate normal distribution

Suppose that $Y_1$ and $Y_2$ follow a bivariate normal distribution with parameters $\mu(Y_1)= \mu(Y_2)= 0, {\sigma^2}(Y_1)= 1, {\sigma^2}(Y_2)= 2$, and $\rho = 1/\sqrt 2$. Find a linear ...
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### How do I find if the probability of the sample proportion is greater than something?

I have this problem and I have no clue how to solve it. In 2012, 31% of the adult population in the US had earned a bachelor’s degree or higher. One hundred people are randomly sampled from the ...
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### Tail difference of quantiles of (symmetric) distribution functions

Assume, for example, $z_\alpha$ are $\Phi^{-1}(\alpha)$ quantiles from standard normal distribution, $\alpha > 0$. If we are interested in the sum$$z_\alpha + z_{1 - \alpha}$$ for standard normal ...
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### Proof and precise formulation of Welch-Satterthwaite equation

In my statistics course notes the Welch-Satterthwaite equation, as used in the derivation of the Welch test, is formulated as follows: Suppose $S_1^2, \ldots, S_n^2$ are sample variances of $n$ ...
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### Correct formula to calculate Chi square statistic [on hold]

I came across two formulas for calculating Chi square statistic. Method-1: Chi square statistic, X²= [(n -1)*s² ]/σ², where n is the sample size, s denotes standard deviation of the sample and σ is ...
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### Probabilities of choosing a red ball:Case 1) 10 distinct balls,8 red and 2 black Case 2) 10balls ,8 identical red and 2 identical black

I know the probability in both cases evaluate out to be 8/10. I want to know the intuition behind solving the probability of choosing a red ball when there are 8 identical red balls and 2 identical ...
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### Tukey's symmetrical lambda distribution

U ~ Uniform(0,1) $$Z_\lambda = \frac{U^\lambda-{(1-U)}^\lambda}{\lambda}$$ I have to find the first four moments and two ($\lambda_1,\lambda_2$) such that they have the same four moments.
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### How to calculate Poisson probability with float $k$?

Ok, so I see the equation is $\frac{λ^k * e^{-λ}}{k!}$ but what if my $k$ is not an integer but a float possibility? for example, $k = 2.5$ and $λ = 5$. I have tried to multiply $k$ and $l$ with the ...
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### Probability of having picked every item from the set at least once after n turns, while picking 3 per turn

Let's say I have a set of 100 items. Each turn, I pick three items at random, note which ones I've picked, and put them back. What is the probability I've picked every item at least once after $n$ ...
I was wondering whether the probability distribution $$P(X = k) = \frac{\lambda}{(1+\lambda)^{k+1}}, \quad k= 0, 1, 2, \dotsc,$$ where $\lambda$ is a fixed positive number, has a name.