Questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution.

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2
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1answer
27 views

Expectation of cumulative distribution function of a standard normal distributed random variable

Let $X$ be a normally distributed random variable with mean $0$ and variance $1$. Let $\Phi$ be the cumulative distribution function of the variable $X$. The find the expectation of $\Phi(X)$. I ...
0
votes
1answer
38 views

How to calculate $E(\sin^2X)$

If $X \sim N(0,1)$ then calculate $E(\sin^2X)$ I understand that $0 < \sin^2x<1$. So the expectation exists. I proceed as $E(\sin^2X)= \int_{-\infty}^{\infty}\sin^2xf(x)\,dx=2 ...
0
votes
1answer
35 views

How to calculate expected value of normal distribution with the condition that value is higher than x

I have following problem. Let assume that lifespan in the population has normal distribution with certain mean, variance and skewness. When the baby is born, its average lifespan will be equal to ...
0
votes
1answer
32 views

Show that for smaller $n$ than $n = 125000$ holds: $\mathbb{P}(|Z_n - \frac{1}{2}| \geqslant 0,01) \leqslant 0,02$

I'm a first year math student and I am having trouble with this exercise: Let $S_n$ be the amount of times we get heads when throwing a coin $n$ times. Let $Z_n = \frac{S_n}{n}$. With the equality ...
1
vote
2answers
44 views

How should I interpret this notation?

I am reading some lecture notes and I'm not sure how to interpret this: $$ b_j(x)=p(x\mid s_j)=N(x;\mu,\sigma^2)$$ It is clear from the context that $N$ refers to normal distribution, but what exactly ...
1
vote
0answers
23 views

Does small perturbation in the denominator explode the expectation of a ratio of two random variables?

The puzzling thing I am facing is Suppose we have two random variables $X$ and $R$ such that $E(X^{-1}R)=1$. Now let $\tilde{X}=X+\mathcal{E}$ where $\mathcal{E}=X\epsilon$ and $\epsilon \sim ...
0
votes
1answer
43 views

Normal distribution for bags of coal produced from a machine.

A machine is used to bag coal, the mass of coal delivered per bag being normally distributed with mean 55 Kg and standard deviation 1.25 Kg. Given two filled bags chosen at random calculate the ...
1
vote
2answers
31 views

Teasing apart an explanation of the Central Limit Theorem

I'm looking at the central limit theorem, and cannot see in the explanation given to me how the average of identical distributions results in the normal distribution. I am told to consider a sequence ...
2
votes
1answer
53 views

What is the best way to interpolate over the 25th and 75th percentile of SAT scores?

The problem: I know the 25th and 75th percentiles of SAT scores for students admitted to a given university, and I want to interpolate over those two points in order to estimate all the percentiles ...
1
vote
2answers
58 views

PDF of $Z=\frac{X^2+Y^2}{2}$ where $X\sim N(0,1)$ and $Y\sim N(0,1)$

Say $X \sim N(0,1)$ and $Y\sim N(0,1)$ are independent random variables. So: $f_X(x) = \frac{1}{\sqrt{2\pi}}e^{\frac{-1}{2}x^2}$ and $f_Y(y) = \frac{1}{\sqrt{2\pi}}e^{\frac{-1}{2}y^2}$. Now I am ...
10
votes
0answers
104 views

Looking for references related to an inequality in order statistics

I was reading the paper "on the minimum of several random variables". In example 10 item (ii) it states: Let $1\leq k\leq n$. Let $g_i,i\leq n$, be independent $N(0,1)$ Gaussian random variables. ...
2
votes
0answers
38 views

Prove or disprove that the Bhattacharyya distance is a true distance function

Let $\mathcal{X}\equiv\Bbb{R}^n\times\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ is the space of all symmetric positive-definite $n\times n$ real matrices. Let $x,y\in\mathcal{X}$, where $$ ...
0
votes
0answers
20 views

Sum of Two Normal Distributions - Weighted?

The problem is given as such: John owns a portfolio with two stocks, ABC and XYZ. He has invested \$400 in ABC and \$600 in XYZ. The quarterly return on ABC is normally distributed with a mean of 7% ...
0
votes
0answers
26 views

Obtaining the standard deviation from a pair of (truncated) normal distributions

The expectation value of one side truncated (upper tail) normal distribution is defined as follows: $$ \operatorname{E}(X \mid X>a) = \mu +\sigma\lambda(\alpha) \!$$ where $$ ...
-1
votes
1answer
29 views

central moment of normal distribution

In the normal distribution mean=2 &variance=4 then, 4th central moment is How to find out 4th central moment I solve through normal variate
1
vote
1answer
55 views

Is this PMF or PDF?

I am reading a technical report on expectation-maximization (EM) algorithm (http://melodi.ee.washington.edu/people/bilmes/mypapers/em.pdf) and I am confused about something. For HMMs, it defines ...
0
votes
1answer
28 views

Finding the mean of normal distribution through integration over $[a, b]$

I know the formula to finding this mean is to integrate $x\frac{1}{b}$ from $a$ to $b$. Can someone explain why this is so? I've been trying to compute the mean with the standard formula ($\int_a^b ...
0
votes
1answer
45 views

For $A,B,C$ independent and normal, what is $I(A+B;\ A+C)$?

Say $A,B,C$ are mutually independent and normally distributed with zero mean but possibly different variances $\sigma_1,\sigma_2,\sigma_3$. What is the mutual information between $A+B$ and $A+C$? All ...
1
vote
2answers
32 views

Central Limit Theorem and Normal Distribution problem.

Suppose I have a sample of people of size $n$ in which the probability that one smokes is p. I am asked what n should be so that the proportion of smokers in the samples is, in approximation of 0.01, ...
1
vote
1answer
37 views

Conditional density, bivariate normal

Let $Z=X+Y$ where $X \sim N(\mu,\sigma^2)$ and $Y \sim N(0,1)$ are independent. What is the conditional density of X given Z, $f_{X|Z}(x|z)$? I already found that ...
1
vote
1answer
14 views

Verification of linear combinations of a normal distribution

A machined part consists of 5 independent components connected end-to-end. Two of these have lengths $N(37.0, 0.49)$, and three of these have lengths $N(24.0, 0.09)$. All measurements are in mm. What ...
1
vote
3answers
48 views

How many tickets can you sell for a plane?

I'm trying to learn about limit theorems, but I have no idea how to calculate the following question: Suppose that 15% of people don’t show up for a flight, and suppose that their decisions are ...
2
votes
1answer
53 views

Covariance between squared and exponential of Gaussian random variables

Assuming the random vector $[X \ \ Y]'$ follows a bivariate Gaussian distribution with mean $[\mu_X \ \ \mu_Y]'$, and covariance matrix $ \left[ \begin{array}{cc} \sigma_X ^2 & \sigma_Y \ \sigma_X ...
2
votes
1answer
25 views

Joint density of normal random variables

Let $Z=X+Y$ where $X$~$N(\mu,\sigma^2)$ and $Y$~$N(0,1)$ are independents. Find the joint density of Z and X. This is the first time I see something like that, look what I did below: I know ...
1
vote
0answers
35 views

Hamming weight in multiple label

Assume you have a $N$ balls. You give each ball $T$ different labels randomly from $\{0,\dots, N-1\}$. So hamming weight of each of labelling varies from $0$ to $\lceil\log_2 N\rceil$. What is ...
0
votes
0answers
32 views

Find the distribution of $Z=\frac{X_1+X_2}{X_1X_2}$, where $X_1$, $X_2$ follow normal distribution

Lets assume $X_1$, $X_2$ follow normal distribution. I am looking for the distribution of: $$Z = \frac{(X_1+X_2)}{X_1*X_2} $$ This is what I have thought so far: The distribution of the ...
-1
votes
1answer
28 views

Probability/Statistics help? [closed]

In a population, height of females are normally distributed with mean $\mu_1 = 162\mathrm{cm}$ and standard deviation $\sigma_1=6\mathrm{cm}$. Heights of males are normally distributed with mean ...
1
vote
0answers
9 views

T-ratio. Estimation of standard error.

Let $ X = (X_1, ..., X_n)$ be a vector observation collected from Normal Distribution. We don't know neither variance of population nor expected value. We would like to estimate expected value for ...
2
votes
1answer
40 views

Standard distribution formulae trick

I am trying to understand the following question. The height of adult males is normally distributed with a mean of 172cm and standard deviation of 8cm. If 99% of adult males exceed a certain height, ...
2
votes
1answer
13 views

Integral estimate: upper bound normal distribution

I'm looking at the proof of Donsker's theorem and it is used that $\frac{1}{\delta}\int_{\frac{1}{\sqrt{\delta}}}^\infty{e^{-y^2/2}}dy\leq \int_{\frac{1}{\sqrt{\delta}}}^\infty{y^2e^{-y^2/2}}dy$ ...
1
vote
2answers
25 views

Proving that the maximum values of these different, Normal distribution, curves are different.

In a question, one variable X is Normally distributed with mean=100, variance=25 and Y is Normally distributed with mean=110, variance=36. The question asks to sketch the p.d.f of each on the same ...
1
vote
2answers
33 views

Sum and difference of three normally distributed variables

We are given three independent random variables $X, Y, Z$ with normal distribution $\mathcal{N}(1,2)$. Are $U=Z-Y+X$ and $V=X+Y$ independent? I thought I would compute the joint density $f_{UV}$ and ...
0
votes
0answers
14 views

Maximizing the weighted sum of two CDFs subject to a constraint on the expected value.

I encountered this problem in a proof and would like to have your help: Consider the maximization problem: \begin{eqnarray} \max_{x,y}b_x\Phi(x)+b_y\Phi(y),s.t\\ ...
2
votes
0answers
29 views

Distribution of $\langle A,x\rangle\langle A,y\rangle + \langle B,x\rangle\langle B,y\rangle$ given $\langle x, y\rangle$

Let $A$ and $B$ be independent, normal distributed $N(0,1)$ normalized unit vectors, and let $x$ and $y$ be unit vectors with given inner product $\langle x, y\rangle=u$. Can we write the ...
1
vote
1answer
45 views

grading on a curve using normal distribution

suppose we have a grade list: $ \text{grades}=\{2,3,5,7,8,10,9,9.75,8,0,11,10,10,3,5.25,13,14,20,18,9\}; $ which mean equals to 8.75 and Standard deviation is 5.06471. we want to improve the grade ...
0
votes
1answer
44 views

statistics and biased estimator of normal distributions

Let $X_1, X_2 , X_3 , X_4$ and be independent, identically distributed random variables from a population with mean $\mu = 10$ and variance $\sigma^2 = 10$ . Let $\bar Y = \frac{1}{4}(X_1 + X_2 + ...
0
votes
0answers
53 views

Normal Exponential Convolution proof

I am seeing a scientific paper where they explain background correction modelled as two random independent variables one with exponential distribution and the other with normal distribution. ...
0
votes
0answers
35 views

Are Gaussians a basis for the vector space of continuous functions?

How can I prove (or disprove) that the Gaussian function family: $f_{\mu,\sigma}(x)=e^{-\frac{(x - \mu)^2}{2 \sigma^2}}$ Are a basis for $C(\mathbb{R})$ ?
1
vote
1answer
33 views

Normally distributed variable with normally distributed mean.

What the idea behind the prove of following statement? I am pretty sure the statement it is correct. If $X \sim N(\text{mean}_x, \text{var}_x)$ and $Y \sim N(\text{mean}_y+X, \text{var}_y)$, then $Y ...
2
votes
0answers
28 views

Solving a nonlinear constrained optimization involving CDF and expectation of normal distribution

I would like to know if it is possible to solve the following nonlinear constrained optimization problem and find how the optimal solution varies with $C$ and $\beta$: $\max_{x,y}\beta ...
0
votes
1answer
30 views

Relationship between univariate normal distribution and multivariate normal distribution

Let $a_1, a_2, a_3$ is column vector and $H = [a_1 a_2 a_3]$. If $a_i$ have standard normal distribution, is this following statement true ? $$ vec(H) = [(a_1)^T (a_2)^T (a_3)^T]^T$$ have multivariate ...
0
votes
0answers
50 views

A Gaussian Divided by a Gaussian Equal to A Gaussian Divided by a Constant

I have a neural-network model in which each neuron is associated with an angle $\theta$. Firing rate as a function of $\theta$ is either a Gaussian or a constant. The claim has been made using this ...
0
votes
1answer
25 views

Pdf of a normal variable accepted with probability dependant on the normal variable

Assume $z$ is a standard normal variable. If $z<0$, then we accept it with probability 1. if $z\ge0$, we accept it with probability $e^{-mz}$, where $m>0$. I'm trying to figure out the pdf of ...
0
votes
1answer
78 views

Variance of a quadratic form

I am considering a variance of two forms: $ R(x) = (x-m)^\top A (x-m) + b^\top (x-m) + c $ $ R'(\Delta) = \Delta^\top A \Delta + b^\top \Delta + c $ where $x$ is a random variable of ...
0
votes
1answer
12 views

Proporties of linear-combinations of normally distributed variables.

At my school, to pass an exam, you'll have to score at least 230 points. The results are normally distributed with $\mu=200$ and $\sigma=20$. If I were to consider 10 students who attends the exam, ...
0
votes
0answers
19 views

Height of a point

I wonder if there is way to find out the height of a specific point x in a normal distribution whenever the standard deviation is not given.?
1
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0answers
33 views

Find the distribution of $Z = 1/X_1 + 1/X_2$

Find the distribution of $Z = 1/X_1 + 1/X_2$, where $X_1$ and $X_2$ follow normal distribution. I have $2$ variables with normal distribution, $X_1$ and $X_2$. How can I find the distribution of: ...
0
votes
0answers
13 views

Exponential Demand Periodic Review

I have exponentially distributed demand data and I am trying to find a formula for an 'order up to level (OUL)' periodic review ordering policy. We are not using a re order point for this policy. ...
1
vote
1answer
40 views

Translate exponential distribution into normal distribution

I have a bunch of inventory management formulas that are supposed to be used with normal distributions, however my demand data fits an exponential distribution. Is there any way to translate the ...
1
vote
2answers
33 views

Proving the $Pr(d>0|a+d=\pi)$ is increasing in $\pi$ when a and d are two independent normal distributions.

I was wondering if it is possible to prove the following (or show false otherwise). Given two independently distributed random variables $a\sim \mathcal{N}(\alpha,\sigma_\alpha^2)$ $d\sim ...