0
votes
0answers
30 views

expectation lognormal and normal

I have two random variables $X\sim N(m_{X},\sigma^2_{X})$ and $Y\sim N(m_{Y},\sigma^2_{Y})$ both normally distributed and they're jointly normally distributed as well with correlation $\rho$. I am now ...
2
votes
2answers
69 views

Log normal distribution - Where am I wrong?

Let $X$ be a R.V whose pdf is given by $$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+ ...
1
vote
2answers
37 views

Integration by parts

Integrate using integration by parts: $F(y) = (y+1)e^{-y}$ Find: Evaluate the $\int_{a=0}^{b=\infty}F(y)\;dy$ using integration by parts. Thus far, I've distributed the $e^y$ term and split ...
2
votes
2answers
45 views

Expected value caluclation

I needed help with this problem Basically I have to show that $E(X^2-3x+2)=E(X^2)-3E(X)+2=\frac{2p^2}{(1-p)^2}$. I know that ...
1
vote
0answers
25 views

Slicing through a cuboid containing spheres, how many are exposed to the surface and what is their combined volume

So I place spheres of radius chosen at random from a normal distribution of known mean and standard deviation in a cub or cuboid at random (not overlapping) until a known density of the entire cube is ...
0
votes
1answer
27 views

Rearranging equation with algebra

I'm having a difficult time showing that the two are equivalent: $2(x_1-\theta)(1+(x_2-\theta)^2)+2(x_2-\theta)(1+(x_1-\theta)^2) = 2(\bar{x}-\theta)(1+(x_1-\theta)(x_2-\theta))$ I have multiplied ...
2
votes
1answer
70 views

Conditional probability explained?

Let $F_A$ be the CDF to the random variable $A$ ( and $B$ another independet rv), how do we get that $P(A+B \le s) = \int_{\mathbb{R}} P(A+B \le s\mid A=x ) \, dF_A(x)$ (This is probably a ...
0
votes
2answers
20 views

Marginal density function question

The question and answer is shown but I don't fully understand the answer for part a. Could someone please explain to me why the integral setup for the marginal density function of y1 is from y1 to 1, ...
0
votes
1answer
39 views

Determine the target weight so that no more than 5% of boxes with normal weight distribution contain less than 500 g [closed]

Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. Suppose a law states that no more than 5% ...
0
votes
2answers
31 views

Probability that the call will be answered at time $t$ is given by $f(t)$. Find the median waiting time for the call.

$$f(t) = \begin{cases} 0 & \text{if $t < 0$ } \\ 0.2e^{-t/5} & \text{if $t\geq 0$} \end{cases}$$. $ $ Find the median waiting time for the call. $ $ I cannot understand ...
1
vote
2answers
105 views

What is the probability of going bankrupt in roulette?

Imagine that the bank has the money $M_1$ and the player has the money $M_2$. The rules are the following: You win with a chance of $\frac{17}{36}$ and lose with $\frac{19}{36}$ each round. Now you ...
2
votes
1answer
32 views

derivation law from the call option formula

i am reading a article about the option pricing. and i got stuck with some typical statement. $C(K)=\int (x-K)^+\mu(dx)$ is given. here, $\mu$ is implied law of asset price and C(K) is the price ...
0
votes
2answers
26 views

Reliability Probability problem

What is the Probability that at least one close path is formed from A to B where each switch has a Probability of close = p and each switch acts independent of the other Proposed Solution Let ...
0
votes
0answers
40 views

Card Shuffling and Convergence in Probability

There are $4n$ cards, and we denote the set of cards with number $4k,k \in \{1,2,\ldots,n\}$ as $S$. The we shuffle the whole cards randomly, which means that each permutation will happen with the ...
5
votes
1answer
89 views

What is the expected value of the number of randomly chosen real numbers between $0$ and $1$ needed to reach a sum of $1$? [duplicate]

My friend told me that the answer to this question was $e$, which intrigued me, but he refused to tell me why. My initial intuition was completely wrong. I thought that since the expected value of ...
0
votes
1answer
21 views

Two notions of conditional expectation

For a randomn variable $Y$ and an event $B$ we can define: $$E(Y \mid B) = \frac{E(1_B\cdot Y)}{P(B)}$$ as the conditional expectation. Now, for a sigma algebra $\mathcal{B}$ and sets $B$ in it you ...
0
votes
2answers
37 views

evaluating an integral with complex exponential (spectral density)

I am having a hard time figuring out how to evaluate this integral from a book that I am reading. Here's the background info but I doubt it's highly relevant to the problem at hand: $X$ is a real ...
1
vote
1answer
25 views

How limits should be understood in large deviation theory

In general we say that a function $f\left(n\right)$ satisfies a large deviation principle if: \begin{equation} \lim_{n\rightarrow\infty}-\frac{1}{n}\ln \left[f\left(n\right)\right]=F \end{equation} ...
4
votes
4answers
92 views

How to integrate: $\int_{0}^{\infty}e^{tx}(x^2e^{-x})/2dx$

I'm working on a few moment generating function problems and I came across: $f(x)=(x^2e^{-x})/2$ for $x>0$, and zero otherwise. Find the mean. The mean is the same as the expected value. If we ...
0
votes
1answer
20 views

What is the joint cumulative distribution function of two separate uniformly distributions ~[0,1]?

I was asked to find a joint distribution for two suppliers where each alone has a uniformly distributed demand~U[0,1], how do I do that? In general is it possible to join in general any two non ...
0
votes
2answers
36 views

How was this integral set up to compute $Pr(X+Y) \geq\frac{\pi}{2}$?

I am trying to understand how to deal with the following type of question given two random variables $X$ and $Y$ that are jointly continuous with some pdf: Here: $f_{X,Y}(x,y) = \left\{ \begin{align} ...
0
votes
1answer
27 views

estimating gaussian like integral

How do we estimate the following kind of integrals - $\int_0^{\infty} e^{-x^r} dx $ For r > 1. I basically need this to find normalization constants for probability measures with densities ...
0
votes
2answers
35 views

gamma function question relating to normal distribution

I'm trying to show that $\Gamma(1/2)=\sqrt\pi$. A hint I've been given is to use a change of variable and then relate it to normal distribution density. However, I'm really confused as to how I would ...
2
votes
1answer
144 views

Sum of two gamma/Erlang random variables $\Gamma(m,\lambda)$ and $\Gamma(n, \mu)$ with integer numbers $m \neq n, \lambda \neq \mu$

The gamma distribution with parameters $m > 0$ and $\lambda > 0$ (denoted $\Gamma(m, \lambda)$) has density function $$f(x) = \frac{\lambda e^{-\lambda x} (\lambda x)^{m - 1}}{\Gamma(m)}, x > ...
1
vote
3answers
53 views

Chebyshev inequality for $n=1$?

Wikipedia suggests that Chebyhev's inequality is only true for $n \ge 2$, but I don't see why we have to exclude the case $n=1$? Is wikipedia right? Chebyshev
0
votes
0answers
309 views

How can I derive the PDF from conditional probabilities?

I have some function $P(i)$ which is the probability of success for an experiment on the $i$th trial. The probability mass function for the first successful trial is: $$PMF(n) = \left( ...
0
votes
0answers
24 views

Identically distributed and same characteristic function

If $X,Y$ are identically distributed random variables, then I know that their characteristic functions $\phi_X$ and $\phi_Y$ are the same. Does the converse also hold?
0
votes
0answers
26 views

How do I convert a Gamma-distributed random variable, $ \Gamma(2,1)$ to Erlang distribution?

I know that a Gamma-distributed random variable with $\alpha$ as an integer can be converted to Erlang distribution but how? and how do I write it's new probability density function?
1
vote
1answer
47 views

Pictures for Expectation

Is there a good way to visualize the formula: $$ E(x) = \int_{0}^{\infty} 1 - F(X) \,\mathrm{d}x $$ ? for positive continuous random variables? I understand the formula as far as basic calculus ...
0
votes
1answer
69 views

Partition a set into n subsets with elements with sum equal to m

For example, given the set $a_n = \frac{n}{20}$ for $\,n \in \{1,\,2,\,\dots,\,19\}$, I would like to get all possible partitions of this set in 4 subsets such that the sum of their elements is always ...
0
votes
1answer
15 views

Mean of cumulative distribution function

Suppose a CDF given by: $F(x)=0$ if $x < -2$, $0.6$ if $-2 <= x <= 1$, $1$ if $x > 1$. How can I then calculate $E[X]$ and $E[X^2]$? I found that $E[X]=0$ since there are only jumps, so ...
4
votes
2answers
234 views

Derivation 9.97 in Jaynes' Probability Theory

In page 298 of Jaynes' Probability Theory: the Logic of Science, equation (9.97), Jaynes says: We expect that, if hypothesis $H$ is true, then $n_k$ will be close to $np_k$, in the sense that the ...
0
votes
1answer
29 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 $-\infty<x<\infty$ ...
1
vote
1answer
29 views

expectation of a standard normal within the CDF

I was wondering if there is a way to compute the following expectation analytically? $ \mathbb{E}\left[e^{cZ}\Phi\left(a+bZ\right)\right] $ where $Z$ is a standard normal random variable and $\Phi$ ...
0
votes
1answer
17 views

Finding the Marginal of $f_y(x,y)$ from the density function

So I have the following function $f(x,y)=10xy^2$ and I am asked to find the marginal of $y$ with the region being $0<x<y<1$. This is my setup: $f_y=\int_y^1 10xy^2 dx$ This is correct right ...
1
vote
1answer
53 views

$Pr(X+Y \geq \frac{\pi}{2})$

I want to find $Pr(X+Y \geq \frac{\pi}{2})$ for joint pdf $f_{X,Y}(x,y) = x \cos y, 0 \lt x \lt \frac{\pi}{2}, 0 \lt y \lt x, 0$ otherwise. I believe I have found conditional pdf of $Y$ given $X=x$ ...
1
vote
1answer
34 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 ...
2
votes
1answer
54 views

Finding the conditional pdf of $Y$ given $X=x$ from a joint pdf. Answer confirmation!

I have a continuous joint pdf, and I am working out the conditional pdf of Y given X=x. Is my method correct? I am given: $f_{X,Y}(x,y) = x\cos y , 0 \lt x \lt \frac{\pi}{2}, 0 \lt y \lt x$ ...
1
vote
1answer
104 views

How to evaluate $\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$

How to evaluate $$\frac{\Gamma\left(\frac{n}{2}\right)}{\Gamma\left(\frac{n-1}{2}\right)}$$, where n is integer > 0? I know the gamma function formula will give $$ ...
0
votes
0answers
17 views

Scaling model output to be between 0 and 1

I have fitted Cox model and the output is generated as: $e^{\beta x}$, where $\beta$ is the coefficient. Now, I would like to have the model output ranging between $0$ and $1$. I'm currently using ...
1
vote
1answer
20 views

Calculate the Marginal Probability

f($X_1$, $X_2$| $p_1$, $p_2$) = $p_1^{x_1}$(1-$p_1)^{({n_1}-{x_1})}$$p_2^{x_2}$(1-$p_2)^{({n_2}-{x_2})}$ $p_1$~Unif(0,1) independently $p_2$~Unif(0,1) $n_1$=34 $n_2$=56 Calculate the marginal ...
0
votes
0answers
34 views

Simplification of Double Integral with Independent Parameters

I am trying to find a posterior distribution and the hint is that the double integral in the denominator should simplify because $p1$ and $p2$ are independent. $\displaystyle \int$$\displaystyle ...
1
vote
1answer
60 views

Two Methods of computing E[X] but I get 2 different answers instead of the same

The 1st method is $\int_{A}^{B}xf(x) dx$ and the 2nd method is $A+\int_{A}^{B} 1-F(X)$ I have the following CDF $$F(X)=\begin{cases} 0\qquad x<2\\ \dfrac{(x-2)^2}{3}+0.3\qquad 2\leq x < 3\\1 ...
1
vote
1answer
39 views

Finding the mean with absolute value

This question is out of my field and topic that I am teaching myself now, but I was wondering how would you solve this problem if it had the absolute value of it. My Question: $$f(x) = ...
3
votes
1answer
31 views

Maximum likelihood to throw exactly two 6s

One throws a dice $n$ times. For which value of $n$ is maximum the probability to obtain exactly two 6s? I get $$n=11 \text{ or } n=12.$$ My solution: the probability to obtain exactly two 6s in ...
0
votes
1answer
24 views

Joint continuous random variable conditional probablity

Okay so here is the problem: The PDF is $$f(x,y)=6(1-x)\;\; 0\leq y \leq x \leq 1$$ The question asks find $P(0<Y<0.25\;|\;X=0.5)$. I approached the problem like this ...
1
vote
1answer
34 views

Weird question about probability density function

I'm assuming "actual" means the total probability of the PDF (the integral from $-\infty to \infty$) must be 1, so $$\int\limits_{-\infty}^{\infty} ke^{-0.1t}dt = 1$$ Wolfram Alpha seems to be ...
0
votes
1answer
47 views

Double Integration with interesting variable limits, and difficult function

I am trying to reconstruct a probabilistic model, I have tried different methods of approach, by parts, substitution, but to no avail. Any help with this would be greatly appreciated!
1
vote
2answers
98 views

Using L'hopital's rule to solve problem.

Show that $$\lim_{x \to 0} \frac{-3x }{e^{x/3}}=0 $$ by L'hopital's rule. I know how to solve this without using L'hopital's rule. I was just reading about this and was wondering can we solve it ...
0
votes
0answers
73 views

MMSE estimate for scalar gaussian & uniform prior

I am trying to analyze the behavior of an MMSE estimator given Guassian measurement with scalar variability on an underlying uniform prior distribution. The measurement is generated according to the ...