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

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3
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1answer
454 views

The PMF of the larger of two numbers selected at random from $1,\dots,12$

Two balls are chosen at random from a box containing 12 balls, numbered 1;2; : : : ;12. Let X be the larger of the two numbers obtained. Compute the PMF of X, if the sampling is done (a) ...
-2
votes
2answers
62 views

Operations on distributions

Say we have two r.v X and Y which are independent and differently distributed ( for e.g X follows a bell curve and Y follows an exponential distribution with parameter $\lambda > 0$ What are the ...
0
votes
0answers
279 views

Formula when drawing from multiple urns (Probability)

I'm trying to work out a general solution for a probability problem, but I can't seem to figure out how to go about with it. I manage to calculate the individual probabilities on a per problem basis, ...
3
votes
1answer
407 views

Can sum of two random variables be uniformly distributed

Say $X$ and $Y$ are two random variables where $X\in\{-\alpha,\alpha\}$, $Y\in\{-\alpha,\alpha\}$ and $Z=X+Y$. Is it possible to find two independent random variables with certain pdf (not necessarily ...
1
vote
2answers
99 views

Probability of defective cogs in a carton

A company that manufactures cogs sells them in cartons of 100. It is historically known that about 1% of the cogs manufactured by the company are defective. How do I find an expression for the ...
1
vote
1answer
54 views

Conditional expectation of conditional sum(Not fully complete)

I have a question: Determine the conditional expectation $\mathrm{E}(A|B)$ for: The number $B(\geq 0)$ of bats that leave a cave at the time of a nuclear explosion has a geometric distribution ...
1
vote
0answers
31 views

Properties of this set of functionals (mixed pairings)

(from the 4th page of http://www.math.toronto.edu/mccann/papers/econ.pdf) Let $X$ be a compact Hausdorff space, and let $\omega$ be a Borel probability measure on $X$. A Borel probability measure, ...
1
vote
0answers
22 views

Continuity Correction with replacement

An urn contains 2 white and 8 red marbles. A marble is drawn from the urn 100 times in succession with replacement. What is the probability of drawing more than 75 red marbles? My attempt: $n=100, ...
0
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0answers
38 views

Normal Distribution while finding sigma

I was reading some things about normal distribution and saw this problem in a text a couple days ago. I know it might be a little advanced for me at the moment, but I was wanted to know if someone can ...
0
votes
2answers
61 views

Trying to understand a probability question

I'm trying to understand a probability question regarding a biased coin, not quite sure how to factor in the biased probability in the question, and also I wanted to make sure the answer is correct, ...
2
votes
2answers
1k views

Simple algorithm for generating Poisson distribution

I found a very simple algorithm that draws values from a Poisson distribution from this project. The algorithm's code in Java is: ...
0
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0answers
40 views

derive an expression for density

I have an expression for a distribution function and have to derive the density function. The distribution function has the form: $$F(x) = N\left[\frac{1}{a}(z * N^{-1}(x) - N^{-1}(p))\right]$$ How ...
0
votes
1answer
32 views

estimating probability of a point sampled from a distribution

Assume we have a d-dimensional vector space.Also, we don't have a parametric distribution function, only we have a set of samples in this space which we assume is sampled from one unknown distribution ...
3
votes
1answer
46 views

Deriving MGF for binomial distribution

Using the definition of the binomial distribution, I obtain that: $$\Psi (t) = (pe^t+q)^n $$ I then compute $\Psi ' = npe^t(pe^t+q)^{n-1}$ I then evaluated this at $\Psi'(0)$ and got ...
-1
votes
1answer
51 views

If speeds of two cars are Normal RV s, what is the distribution of the distance between them?

The speeds of two cars are random variables that follow $N(\mu_1,\sigma_1)$ and $N(\mu_2,\sigma_2)$ distributions.They start simultaneously. Let X be the distance between them after m hours. (Note ...
1
vote
2answers
34 views

Distrubution of the autoregressive process

Consider stationary autoregression AR(1): $$u_t=\beta u_{t-1}+ \varepsilon_t, \quad t \in \mathbb{Z}.$$ $\{\varepsilon_t\}$ - i.i.d. $N(0,\sigma^2)$ random variables. I know that $\mathbf{E}u_t =0$ ...
1
vote
2answers
65 views

Find $[E(Y^2)]$. When $Y = 3 * X - 5$ and $X$ is distributed in range $[0, 5]$

Need help with exercise. Random variable $X$ is evenly distributed in range $[0, 5]$. Need to find $E[Y^2]$ when $Y = 3X - 5$ Every hint/tip will be appreciated. Thank you Alternative solution ...
1
vote
1answer
56 views

How to calculate joint probability distribution for replacement sample?

If 3 cards drawn with replacement from 12 face card and $X$ represents the no. of kings` and $Y$ represents no of jack construct joint probability distribution. What I am doing; 12 Face cards having ...
0
votes
1answer
105 views

Random Variable depending on another Random Variable?

Here is a quesiton that I was also able to found on the Internet, here. Actually, I've solved 4 of 5 questions, so I only show you the question that I could not. A Computer system carries out ...
0
votes
0answers
135 views

How to draw contour lines for a bivariate Gaussian distribution by hand!

Suppose that we have calculated eigenvalues and obtained eigenvectors of a bivariate Gaussian distribution, how could we draw it's contour lines by hand? Note that We have a eigenspace (set of ...
1
vote
1answer
44 views

Poisson Exponential Distribution with slight change

How would I derive the CDF of P(T$\ge$t). I know the CDF of an exponential poisson distribution is F(t) = P(T$\le$t) = 1-P(T$>$t) = 1-P(X=0) = 1-$e^{(-\lambda)(t)}$, but in the case of (T$\ge$t) we ...
1
vote
1answer
44 views

Moment generating function of a random number of IIDs?

Let's imagine I've got $Q_1,\ldots,Q_L$ independent identically distributed normal random variables with parameters $\mu, \sigma^2$, and $L$ is binomial with parameters $n,p$. Let $Y=Q_1+\cdots+Q_L$. ...
3
votes
1answer
100 views

if $X$ and $Y$ are i.i.d., and if $X+Y$ and $X-Y$ are independent, are $X$ and $Y$ normally distributed?

Just recently come across Normal Distribution, and the following statement seems to be quite true, but is it? Can someone provide some general proof sketch if so please: For X and Y identically and ...
0
votes
1answer
36 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
3answers
664 views

Find the probability distribution for the number of spades.

Three cards are drawn in succession from a deck without replacement. Find the probability distribution for the number of spades.
1
vote
1answer
103 views

Interpretation of the slope in a probability distribution.

When considering a certain probability-distribution $P(t)$, then what information does the derivative $\frac{\mathrm d P(t)}{\mathrm{dt}}$ give, in general? Context: I raised this question on ...
1
vote
1answer
110 views

Showing the normal distribution has points of inflections at $x = \mu \pm \sigma$ and a maximum at $x = \mu$

$X \sim N(\mu, \sigma^2)$ I.e. the density of $X$ is the normal distribution. I am looking to show that $f_X(x)$ has points of inflections at $x = \mu \pm \sigma$. In my notes it says that we ...
1
vote
1answer
33 views

Two Uniform Independent Random Variables: When is one greater?

You have two independent random variables: $X$ and $Y$, which are both uniformly distributed over $(0,1)$. Consider the inequality $X^2- 4Y < 0$. What percentage of the time is the inequality ...
1
vote
0answers
17 views

What's the distribution of the difference of two gamma distributions? (the two gamma have same shape parameters)

If $X∼\textrm{Gamma}(a,b_1)$ and $Y∼\textrm{Gamma}(a,b_2)$, $X$ and $Y$ are independent, what is the distribution of $X-Y$?
0
votes
1answer
45 views

Strict Inequality Using CDF

Considering the definition of cumulative distribution function: $$F_{x}(x)=P[X\le x]=\int_{- \infty}^{x} f_{x}(x)dx$$ where $f_{x}$ is the probability density function of $x$, how can one obtain ...
2
votes
1answer
43 views

Joint distribution of $X+Y$ and $\frac{X}{X+Y}$

Let $X$ and $Y$ be two random variables i.i.d $U(0,1)$. Find the joint pdf of $T = X+Y$ and $U = \frac{X}{X+Y}$ and the marginal densities of $T$ and $U$ My attempt: We will have the following ...
4
votes
1answer
110 views

Tricky probability problem

I am having trouble with proving the following assertion: $X,Y$ are i.i.d. with mean $0$ and variance $1$. If $X+Y$ and $X-Y$ are independent then $X,Y$ are normally distributed. Should I be ...
5
votes
1answer
328 views

How do you estimate the mean of a Poisson distribution from data?

I have thought of three different approaches for estimating the mean for a Poisson, but I am not sure which one is the correct method to estimate it (the third one is documented separately at the end ...
1
vote
2answers
51 views

Standard Uniform Distribution

I am trying to show that a random variable $X_2$ has a standard uniform distribution. I have: $\alpha \subset(0,1), X_1 \sim U[0,1],$ and $X_2 = \begin{cases} X_1 &\mbox{if } X_1< \alpha ...
1
vote
1answer
73 views

systematic way of finding the bounds for change of variables (multivariable case), Jacobian

Let's say that $X,Y$ are independent standard normal random variables. I am interested in the distribution $P(X+Y\le 2t)$. Clearly, the domain of integration in this case is $-\infty<x<\infty$ ...
2
votes
1answer
112 views

Is Wikipedia correct?

Cantelli's Theorem Wikipedia says: $$P[X-\mu \geq a] \leq \frac{\sigma^2}{\sigma^2+a^2}$$ for $a > 0$ and $$P[X-\mu\geq a] \geq 1- \frac{\sigma^2}{\sigma^2+a^2}$$ for $a <0$. Is the second ...
1
vote
1answer
33 views

Max of symetric random walk on Z

I can't get to solve this, could someone help me? Let $(X_i)$ be such as $P(X_i=1)=P(X_i=-1)= 0.5$ for all $i$ integer such as $1\le i\le n$. Let $S_n=X_1+...+X_n$. Let's now consider $2^n$ ...
1
vote
1answer
200 views

Modelling problem as a Poisson distribution

I'm trying to understand how to model certain problems as a Poisson problem. I'm unable to get the right thinking in place to understand such problems. For example, something like this: Arrivals at ...
1
vote
1answer
106 views

Computing P(X+Y>0) for the joint pdf of X and Y.

Let X and Y be two jointly continuous random variables with the given joint PDF; $\begin{equation} \nonumber f_{XY}(x,y) = \left\{ \begin{array}{l l} \frac{1}{3}x^3+\frac{1}{5}y^2 & ...
0
votes
1answer
13 views

conditional on X=x Y is binomial(X,p), What is E(exp(y)/X=x)

Suppose we know conditional on X=x Y is binomial(x,p) where p is known.What is E(exp(y)/X=x) where exp is the expodential function and E the expectation Any help will be appreciated Thank, You
0
votes
1answer
33 views

Continuous Distributions - Normal Approximation

Suppose 80% of people who buy a new car say they are satisfied with the car when surveyed one year after purchase. Let X be the number of people in a group of 60 randomly chosen new car owners who ...
1
vote
0answers
50 views

multivariate p.d.f. and distribution

Suppose $X\sim N(\mu,V)$ where $\mu = \begin{pmatrix} 2 \\ 2 \\ 2 \end{pmatrix}$ $V = \begin{pmatrix} 3 & 2 & 1 \\ 2& 4 & 1 \\ 1 & 1 & 2 \end{pmatrix}$ a) ...
0
votes
1answer
62 views

Find unbiased for $\theta$ of $\hat{\theta_1} = (n+1) ~y_{(1)}$given a uniform distribution on the interval $[0, \theta]$.

Show that $\hat{\theta_1} = (n+1) ~y_{(1)}$ is unbiased for $\theta$. For $$P[Y_i \le Y] = 1 - y/\theta$$ Then for $ P[Y_{(1)} < y] = 1 - [1-F_{(Y_i)} (y)]^n$ which should equal to $1 - ...
0
votes
1answer
28 views

Use $2\sum_{i=1}^n Y_i/\beta$ which is a pivotal quantity to derive a 95% confidence interval for $\beta$

Suppose $Y_1$,...$Y_n$ is a random sample from a gamma distribution with $\alpha = 2$ and unknown $\beta$. GOAL: Use $2\sum_{i=1}^n Y_i/\beta$ which is a pivotal quantity to derive a 95% confidence ...
1
vote
1answer
38 views

Finding conditionally expected $y$ given a specific $x$ from a joint distribution function!

I want to determine expected $y$, given $x=2$ given joint pdf shown below $$\frac{1}{2y} * e^{-\frac{y^2 + \frac{x}{2}}{y}}$$ for $x,y \gt 0$ and $0$ otherwise I believe this means I want ...
1
vote
1answer
130 views

Sufficient condition for monotone likelihood ratio property

What are sufficient conditions for the monotone likelihood ratio property? I have a set-up where $F(x)$ (cumulative distribution function of r.v. $x$) always exceeds $G(x)$ (a different cum. distrib. ...
0
votes
1answer
69 views

Error Term of Chebyshev inequality?

Chebyshev inequality tells us that $$Pr[|X-E[X]|\geq a]\leq \frac{Var[X]^2}{a^2}$$ Do you know an Expression (or a paper where this Expression is mentioned) for the error term?
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1answer
60 views

A Diminishing Geometric Distribution

A standard geometric distribution can be interpreted as the number of Bernoulli trials required to get one success. However, what if the probability of success if each trial diminishes by some factor ...
1
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1answer
132 views

Calculating expectation conditioned on a sigma algebra

Let $\Omega=(0,\infty)$ and $\mathcal F=\mathcal{B}(\Omega)$. Let $\mathbb P$ be the probability measure corresponding to the exponential distribution with parameter $\lambda$. $X$ and $Y$ are two ...
1
vote
1answer
50 views

Length of holding interval in renewal theory

I have a Poisson process $(N_t)_{t\ge 0}$ of rate $\lambda$. This process refers to the number of renewals up to time $t$. I also define $S_{N_t}$ which refers to the length of the holding interval ...