1
vote
1answer
7 views

Probability mass function of a degenerate distribution

Wikipedia article "degenerate distribution" states that "The probability mass function does not exist." Is it really right? Why can't it be set as $$ f(x) = \begin{cases} 1 & \quad x = x_0, \\ ...
2
votes
2answers
21 views

Bivariate distribution with normal conditions

Define the joint pdf of $(X,Y)$ as: $$f(x,y)\propto \exp(-1/2[Ax^2y^2+x^2+y^2-2Bxy-2Cx-Dy]),$$ where $A,B,C,D$ are constants. Show that the distribution of $X\mid Y=y$ is normal with mean ...
0
votes
0answers
22 views

concentration of the following random variable: number of items that fit in

This is related to this previous question. Let us assume that we have a capacity $n>0$ which tends to infinity. We are given an i.i.d. sequence of nonnegative random variables ...
1
vote
0answers
9 views

Bounds on functions involving CDFs or Beta function

I have functions of the form \begin{align} I_i = \int_0^\infty F_0(x)^aF_1(x)^b(1-F_0(x))^c(1-F_1(x))^ddF_i(x)~~~~i = 0,1 \end{align} $F_0(x)$ and $F_1(x)$ are CDFs corresponding to the random ...
1
vote
1answer
12 views

What is the relation between fix points of random uniform permuation, and probability of independent events occuring.

Let $A_1,\dots,A_n$ be independent events that occur with probability $1/n$ each. Let $p_{n,k}=P($exactly $k$ events occur). One can show with stirlings formula that ...
0
votes
0answers
23 views

Maximizing P{X=Y} where X and Y are Binomial

X~Binomial(N = 100, p=0.5) Y~Binomial(N = 120, p=0.5) What is the largest possible numerical value of P{X=Y}. X and Y are not necessarily independent.
0
votes
0answers
11 views

Joint density function of exponential and gamma distribution

My problem is: $X_1,...,X_n$ are independent exponentially distributed random variables with $\lambda=1$ paremeters. I have to find the joint density funcitions of $ Y=\sum\limits_{i=1}^n{X_i}$ ...
1
vote
1answer
26 views

What is the variance of this random variable: number of items

Let us assume that we have a capacity $n$ which tends to infinity. We have an infinite number of random variables $X_1, X_2, \dotsc$, where each $X_i$ is independent and identically distributed with ...
-2
votes
0answers
10 views

Relation between Bernoulli RV, binomial RV, geometric RV and Poisson RV [on hold]

what is the relation between Bernoulli RV, binomial RV, geometric RV and Poisson RV? And how we represent them?
-2
votes
0answers
25 views

Trouble with Conditional probability and expectation [on hold]

I have a few questions in probability that have been bothering me. The first is this: Why is $$E(T-t | T \ge t) = \int_t^\infty \frac{(s-t)f(s)~ds}{P(T\ge t)}. $$ The second is this: How does one ...
0
votes
0answers
12 views

Help in finding the functional form of the probability density function

This may seem trivial but I will appreciate help in determining the functional form of the probability density function (pdf) for the following case. Will highly appreciate some guidelines on how to ...
2
votes
1answer
13 views

Beta/Dirichlet question

A generalization of the beta distribution is the Dirichlet distribution. In its bi-variate version, (X,Y) have pdf $f(x,y) = Cx^{a-1}y^{b-1}(1-x-y)^{c-1}, 0<x<1, 0<y<1, ...
1
vote
2answers
32 views

Probability of Snow in New York

In New York, snow is reported 25% of days in February. If this trend continues, what is the probability that it will snow exactly 9 days this coming February and is not a leap year? Solve this ...
0
votes
0answers
22 views

cantor staircase function uniform distribution on cantor set

suppose Cantor staircase function $F$ is extended to have $F(a)=0$ for $a<0$ and $F(a)=1$ for $a>1$. Then how can one show that $F$ is the cumulative distribution function of the uniform ...
0
votes
0answers
21 views

Finding the roots of a polinomial function obtained by a Binomial c.d.f.

I came across with the following question and I am also attempting to solve it. Let $B(K/2;K,1-x)$ be the Binomial c.d.f. with $K$ trials having at least $K/2$ success with each trial having success ...
2
votes
1answer
43 views

How to calculate conditional probability with inequality

I know that: \begin{equation}\displaystyle P(A=x|A+B=y) = \frac{P(A=x \cap A+B=y)}{P(A+B=y)}\end{equation} Assuming $A$ and $B$ are independent, the intersection of the two events can be resolved as ...
0
votes
1answer
33 views

Z~U[0,1] and X=f(Z) and f is:

I have found the f(z): Now, I need to find pdf of X. And I can see that 0< f(Z)=X<1, I don't know how I am going to get f(X), I just can see that f(X)=0 when X<0 and x>1, but I can see a ...
0
votes
2answers
17 views

Simple finding the PDF given function

I am a little confused on how to go about finding the PDF given a condition for a function. So I have the function $$ Y(x)=ae^{-bx} \,\,\,\,\,\,\, a,b,x \geq0 $$ and I need to find the value for X ...
0
votes
0answers
12 views

approximation using a shifted gamma

I have a PDF that is weighted sum of a gamma and shifted gamma distribution f(x)=0.75*gamma(x,100,0.1)+0.25*gamma(x-10,1,10) is it possible to approximate this PDF by a shifted gamma that has the ...
1
vote
1answer
40 views

Central limit theorem kind of statement for records

I am trying to prove the following statement, but I do not know how to go on: Let $F(x)$ be an arbitrary continuous distribution function. Then there are constants $A_n, B_n > 0$ such that, as ...
0
votes
0answers
9 views

A cell dies at a constant rate r and what is the density function of its life time.

The problem: A cell dies at a constant rate $r$ and the its life time is the duration from t=0 to when when it dies. what is the density function of its life time $l$? I have done some relevant ...
0
votes
1answer
13 views

Using a joint distribution table to find probability?

I have the following joint distribution table. I am trying to answer the following questions. A,B,C,D For (a) I put $P(X=1, Y=2)=1/20$ (B) $p(x=0,1\le y<3)= 1/4+1/8$ But I am not sure how to ...
3
votes
0answers
34 views

Distribution of $\frac{X}{|Y|}$, where X and Y are standard normal r.v.'s

Let X and Y be independent standard normal random variables. What is the distribution of $\large \frac{X}{|Y|}$? Attempt: Let $\large U = \frac{X}{|Y|}$ and $ V = |Y|$. This transformation is not ...
0
votes
1answer
23 views

Probability question using Poisson

Here is my Question: A country bus driver picks up passengers randomly and independently at a mean rate of 12 per hour. (i)Find, correct to 3 decimal places, the probability that he picks up ...
1
vote
1answer
35 views

Joint density calculation-Spot the error

Suppose $X_1 $ and $X_2$ are i.i.d standard normal r.v.s and $Y=X_1^2+X_2^2$, then we know $Y \sim \chi_2^2 $ and $f_Y(y)= \frac{1}{2}e^{\frac{-y}{2}}$. Using the identity $f_{X,Y}=f_{X\mid Y} \cdot ...
0
votes
1answer
29 views

Binomial Distribution Proof

What is that $I(\cdot)$ in the 3rd step means? $p_{x_n}(y_n-y_{n-1}) = p(X_n=y_n-y_{n-1}) = p(X_n)$ belongs to the interval $\{0,1\}$, since it is a random variable. Then, how are getting to that ...
0
votes
1answer
26 views

Bernoulli distribution solving for n

So we have this missile protection system that has $n$ radar sets that are all independent. Each have a probability of $0.9$ of detecting a missile. How large must $n$ be if we want the probability ...
0
votes
2answers
26 views

Probability to pass multiple-choice test, with two type of questions

First i want to say there are a lot of questions related to this, but i couldn't find a similar case. Suppose we have the typical problem where we need to compute the probability of pass a ...
-1
votes
0answers
15 views

Sum of the smallest or greatest k components of a random vector drawn from a symmetric Dirichlet distribution? [closed]

Is any distribution known for the sum of the smallest or greatest $k$ components of a random vector drawn from a symmetric Dirichlet distribution?
1
vote
1answer
15 views

Splitting intervals with cut-off

I would like determine the cumulative distribution function (cdf) of the following random-variable X: Suppose we have the following process: The unit interval is split into two pieces at a point $u$, ...
0
votes
1answer
17 views

How can I find the expected value of a random variable with terms that increase until infinity?

Here is the question A company buys a policy to insure its revenue in the event of major snow storms that shut down business. The policy pays nothing for the first such snowstorm of the year and ...
0
votes
1answer
31 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
0answers
18 views

Confidence interval for the conversion on site

I am the developer of web service and I'm trying to to build some plots for the inner dashboard. I raised two questions that I can not solve on their own. Suppose ...
1
vote
3answers
31 views

Find the $P(X>1)$ for the given pdf?

A part of this question asks me to find the $\Pr(X>1)$ given that $$f_X(x) = \begin{cases}\frac{1}{\sqrt{4x}} & 1 <x<4 \cr 0 & \text{otherwise} \end{cases}$$ I solved this by taking ...
-3
votes
2answers
21 views

probability distributions ! what is the solution [closed]

among group of 500 students 30% are males 10% of males and 20% of females are left handed one student is selected at random what is the prob that the selected student is left handed ? help me please
0
votes
1answer
25 views

Convergence in distribution and probability

Suppose ${X_{n}}$ is a sequence of non-negative random variables with cumulative distribution function given by $F_{X_{n}}(x) = 1 - 1/(1+nx)$ for $x\geq 0$. Examine if $\{X_{n}\}$ converges in ...
0
votes
1answer
13 views

Finding the probability of a probability density function

Suppose that $f(x) = e^{−x}$ for $0 < x$. find $P(1 < X)$ I know typically we integrate $f(x)$ from $1$ to $\infty$ but in this case $x = 1$ is not included, how do I go about doing this? All ...
2
votes
1answer
43 views

Joint distribution of range $(R=X_n-X_1)$ and mid-range $(V=\frac{1}{2}(X_1+X_n)$order statistics

Let $X_1,X_2, · · · , X_n$ be independent and identically distributed Uniform random variables on the interval (0, a) for a > 0, each having a density function $f(x) = \frac{1}{a}$, $0<x<a$. Let ...
-1
votes
1answer
40 views

Probability: Random Variables and Probability Distributions

1) The function: $F(x)=k(1-(1/2)^{[x]})$, $x > 0$ Is the distribution function for a discrete random variable X. Here, [x] denotes the integer part of x (i.e., the greatest integer less than or ...
1
vote
0answers
29 views

$U$-Uniform$(0,2\pi)$, Z-Exp$(1)$, $U$ and $Z$ are independent. Then $\sqrt{2Z}\cos U$ and $\sqrt{2Z}\sin U$ are independent standard normal. [closed]

Given that $U$-Uniform$(0,2\pi)$, Z-Exp($1$), $U$ and $Z$ are independent. Show that $\sqrt{2Z}\cos U$ and $\sqrt{2Z}\sin U$ are independent standard normal variables. Thanks in advance for any ...
2
votes
1answer
23 views

Exchangeable/Independent Bernoulli Distribution

Let P be a uniform random variable on the interval $(0,1)$ with density function f(p) = 1, $0<p<1$. Let $X_i|P$, i = 1,2,...,n be independent and identically distributed random variables having ...
1
vote
1answer
16 views

Travelling from one destination to another

This is the problem : Manish has to travel from A to D changing buses at stops B and C enroute. The maximum waiting time at either stop can be 8 minutes each, but any time of waiting up to 8 minutes ...
0
votes
1answer
31 views

Probability of sample mean [closed]

A town has $500$ real estate agents. The mean value of the properties sold in a year by these agents is $\$800,000$ and the standard deviation is $\$300,000$. A random sample of $100$ agents is ...
1
vote
1answer
45 views

Apparently same probability questions with different answers.

I was reading A first course in probability by Sheldon Ross when and then I came up with this question. This is how he introduces the famous problem of points Independent trails, resulting in a ...
0
votes
0answers
53 views

Probability and Expected profit

I really need help for this qn! You are asked to determine the profitability of a new line of sunglasses, which will retail for \$10. The fixed cost of setting up the line is \$2000. The total number ...
0
votes
2answers
32 views

Flipping several biased coins

Assuming I'm flipping $M$ biased coins with different probability for heads $p_i, i=\{1,...,M\}$. What is the probability of having $k$ times head? Is there a distribution function known for this?
0
votes
2answers
40 views

How can I do a constructive proof of this:

Say Z is a non-negative R.V, and P(Z>0)>0. Then exists a a>0 and an b>0 such P(Z>a)>b. I am not sure how to start with the proof, I've been assigning numbers than can qualify for some CDFs but I don´t ...
0
votes
0answers
46 views

Make the sum of random variables converge, while the sum of the variances diverges [closed]

Suppose $X_n$, $n=1,2,3,...$, are independent and $Var(X_n)$ is uniformly bounded by finite constant $C>0$. Construct $X_n$ such that $\sum_nX_n$ converges a.s., but $\sum_nVar(X_n)=\infty$.
0
votes
1answer
32 views

Pdf of the product of an exponential r.v. and a beta r.v.

Let $X$ and $Y$ are 2 independent random variables, where $X$ has an exponential distribution with parameter $1$ and $Y$ is $\beta(a,b)$ distributed. What is the Pdf of $W=XY$ ? Thanks !
1
vote
1answer
28 views

Ratio between normal distributed and gamma distributed variables

Let $X \sim N(0,1)$ and $G \sim Gamma(a)$. Why is $\frac{X}{G}$ t-distributed?