For questions involving random variables uniformly distributed on a subset of a measure space. To be used with [probability] or [probability-theory] tag.

# Questions tagged [uniform-distribution]

2209 questions

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### Why is Y uniform when Y is equal to the distribution function of X?

If X has distribution function F(x) of the continuous type such F(x) is strictly increasing when 0 < F(x) < 1 (some say continuity is enough?). Then Y = F(x) follows uniform over 0 & 1?
Why is this true? I proved it but I'm struggling with the…

Kunal Garg

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### Find $P(X+Y > \frac{3}{2})$ given that X is uniformly distributed on [0,1] and the conditional distribution of Y given X=x is uniform on [1-x,1].

Suppose $X$ has the uniform distribution on the interval [0,1] and that the distribution of random variable $Y$ given that $X=x$ is uniform on the interval $[1-x,1].$
Find the probability that $P(X+Y > \frac{3}{2})$ given to 4 decimal places.
So far…

rl4y

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### $X$ and $Y$ are $\text{unif}(0,1)$. Find the Joint Distribution of $S$ and $U$, where $S=X+Y$, $U=\min(X,Y)$

$X$ and $Y$ are $\text{unif}(0,1)$. Find the Joint Distribution of $S$ and $U$, where $S=X+Y$, $U=\min(X,Y)$.
I know the density of $S$ is $f(s)=s$ for $0~~
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Charlottee

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### Give example of 2 standard uniform random variables with given Pearson correlation

Give example of r.v. $X$ and $Y$ with $X \sim U(0,1)$ and $Y \sim U(0,1)$ and $corr(X,Y) = .25$.
How does one approach such a problem?

Jeff Nguyen

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### Exponential and Uniform Distribution Problems

Questions
I have the following problems on my probability module:
Qu1
Suppose that X is uniformly distributed over $[−3, 3]$.
(a) Find the probability that the first digit of $X$ after the decimal point is $3$.
(b) Find the distribution function of…

George Cooper

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### $(X,Y) \sim U[0,1]^2, (X - Y), (X+Y) \sim $ what distribution?

$(X,Y) \sim U [0,1]^2$. Therefore $X,Y \sim U[0,1]$ each and $X$ and $Y$ are independent. What are the distributions of $X-Y$ and $X+Y$?
My approach: It is my intuitive understanding that $X-Y$ should follow a symmetric triangular distribution over…

Canine360

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### Finding probability density function for $|2M+1|$, with $M \in [-1,4]$

Let $M$ be a number randomly selected from $[-1,4]$. Find the density function of $N=|2M+1|$.
My Attempt: I'll use the method of distribution functions.
First I note that
$(i)$ $N = -2X-1 : X \in[-1,\frac{-1}{2}) \Rightarrow Y \in (0,1]$
$(ii)$ $N=…

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### What's the distribution of the random variable ${F_X}^{-1}(U)$ where $U$ is a uniform random variable on $[0,1]$

Let $X$ be a continuos random variable taking values in $[a,b]$ with c.d.f $F_X$ which is strictly increasing on $[a,b]
(a) Show that the random variable $F_X(X)$ has a uniform distribution on $[0,1]$.
(b) Let $U$ be a uniform random variable on…

Andrey

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### Sample mean converges to 1/2 with error less than 0.01 and probability 0.99

The sample mean random variable of $N$ IID random variables with $X_i$ ~ $U(0, 1)$ will converge to 1/2. How many random variables need to be averaged before we can assert that the approximate probability of an error of not more than 0.01 in…

darisoy

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### Distribution of the Sum of Uniform Random Variable and Another Random Variable

This could have been asked somewhere else, but I couldn't find it anywhere. Assume $X$ and $Y$ are random variables which take values in $\mathbb{Z}_n$ - i.e., $\{0, 1, 2, ..., n-1\}$. Also assume that $X$ is uniformly distributed. What is the…

John

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### Determine the posterior density of $θ$ after doing $n$ coin tosses

I am given a that I toss a coin and the prior $θ \stackrel{}{\sim} Uniform[0.4, 0.6]$
$\textbf{Note:}$ $\theta$ is the probability of getting a head on a single toss
If I toss the coin $n$ times and obtain $n$ heads, then what is the posterior…

VirgOpta

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### Convert a uniform distribution to two uniform distributions

I have a distribution function:
$$ F(x) = \begin{cases} 0 & x \leqslant 0, \\ \frac{x}{2} & 0 < x \leqslant 1 \\ \frac{2}{3} & 1 < x \leqslant 2 \\ \frac{2}{3} + \frac{x-2}{3} & 2 < x \leqslant 3, & \\ 1 & x > 3 \end{cases} $$
I want to find such $…

Vladislav Kharlamov

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### Probability Theory - Independent Uniform Variables

Suppose $X, Y$ are independent uniform(0,1) random variables. For some arbitrary $t$, I want to find $P(X/Y \le t)$. I am trying to draw a picture to figure this out, but I don't think I am accomplishing too much. Could I have some direction on how…

324

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### Posterior Probability Distribution from Geometric Distribution

Suppose an experimenter believes that $p \sim Unif([0, 1])$ is the chance of success on any given experiment. The experimenter then tries the experiment over and over until the experiment is a success. This number of trials will be $[G \mid p]…

Julia

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### Conditional expectation (uniform distribution)

We have $X_1, X_2, \ldots$ - independent random variables with uniform distribution on interval $[0,1]$. $N$ - random variable independent of $X_1,X_2,\ldots$ with Poisson distribution with $\lambda=2$. I have to calculate $\operatorname{cov}({V_N,…

HSmile

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