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

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61 views

Integral of normal distribution curve

I am having hoping to use the integral of the normal distribution curve to find the probability of having a mean of $0.30$ or greater, i.e. one tailed distribution. With a sample standard deviation of ...
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0answers
16 views

Chi sqaured table for degrees of freedom 616?

In order to check heteroskedasticity, we use the White's test. I tried to follow this method below, however, could not find a table with df=2016 and 95,5% confidence. I don't understand how we get ...
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0answers
17 views

Product of two gaussion functions [duplicate]

I would like to ask how to calculate the product of two Gaussion functions? As far as I have understand the product is another Gaussion function. How can this be? I wonder if the std.dev. will be ...
0
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0answers
27 views

Determination of a Lower Confidence Bound

Two stochastic variables $X$ & $Y$are normally distributed with $X\sim \mathcal{N} (0,1)$ and $Y\sim \mathcal{N} (2,1)$. We declare that $W = X + Y$ and $W$ is normally distributed with $W\sim ...
1
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1answer
44 views

Standard Normal Distribution Transformation Z=lnY

I'm not sure if my approach to this problem is correct and I need help I need to apply $Z=\ln{Y}$ to the following standard normal distribution and then find the distribution of $Y$ ...
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1answer
31 views
2
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1answer
54 views

Numerical Approximations to the Cumulative Distribution Function of the Normal Distribution

I have been trying to write the code for the Cumulative Distribution function (CDF) of the normal distribution in C++. Since the cdf does not have a closed form solution of the integral, I was ...
2
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2answers
39 views

Probability that $x > 0$ given that $x > y$ for independent and normally distributed $x,y$

I was recently been asked this question in an interview but not able to solve it as I am rusted in Bayesian conditional probability. Here is the question: $x$ and $y$ are independent variables ...
2
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1answer
35 views

Why is the $0$th percentile of the standard normal distribution $-\infty$?

Why is the $0$th percentile of the standard normal distribution $-\infty$? I can't explain the cause except saying there is no area under the curve. So it goes beyond the bell-shaped curve.
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1answer
43 views

Independence of Gaussian Distribution Function with Different Means

Is there any way to prove that $N$ Gaussian distribution functions are linearly independent if and only if the means are different. For example, if ...
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0answers
36 views

Standard Deviation of Binomial vs Sampling Distribution

High school dropouts make up 14.1% of all Americans aged 18 to 24. A vocational school that wants to attract dropouts mails an advertising flyer to 25,000 persons between the ages of 18 and 24. (a) If ...
1
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1answer
63 views

How to derive the expected value of even powers of a standard normal random variable?

I am trying to prove that, for a standard normal random variable $Z \sim N(0,1)$, ${\mathbb E}[z^n]=n!!$ for even values of $n$. What I'm doing is integrating the p.d.f. of $Z$ which is ...
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2answers
21 views

How to analytically find probability of certain combinations between normally distributed populations?

Let's say we have two normally distributed populations A and B, which have different means and standard deviations. We then pick one item from population A and one from population B. How could we ...
2
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1answer
31 views

Is the following process bounded (iterative normal sampling)

We define the following stochastic process: $X_0=1$ $\forall i\geq i:X_i\sim\mathcal N(0,X_{i-1}^2)$ That is, we first sample $X_1$ from the normal distribution with variance $1$, then in the ...
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0answers
19 views

Effect of Normalization On a Distribution

Let's assume that we have calculated some values proportional to probabilities for a special distribution(some hypothetical distribution other than normal distribution but we don't know which ...
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0answers
21 views

Posterior of mean given an observation from a bivariate normal with unknown but common mean, and known variance

suppose the sample vector $(x,y)$ is generated from a bivariate normal: $$ \left[\begin{array}{c} x\\ y \end{array}\right]\sim N\left(\left[\begin{array}{c} \theta\\ \theta ...
1
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1answer
95 views

An integral with density function of $N(\hat{a}, \frac{1}{s})$

I am stucked on this integral, which is from a research paper in Finance, for a while, so can anyone please help walk me through how we can get the answer on the RHS of this integral? Prove: ...
0
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1answer
26 views

$P(X=c)=0$ for normally distributed $X$?

Let $X$ be norm $(a, b)$-distributed and let $c$ be some real number. Does this imply $P(X=c)=0$? What if $b=0$?
1
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1answer
74 views

$G_n:=\sqrt{n} \left(X_n-1\right) \xrightarrow[n]{d} N(\mu,\sigma^2) $ implies $\sqrt{n} \left(1-X_n^{-1}\right)=G_n+o_P(1)$

Let $X_n$ be a sequence of RV so that $G_n:=\sqrt{n} \left(X_n-1\right) \underset{n \to \infty}{\overset{d}{\longrightarrow}} G \sim N(\mu,\sigma^2)$. I want to show that in this case $\sqrt{n} ...
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0answers
21 views

Iso-density locus of Gaussian mixture distribution

I would like to known what is the equation of the iso-density surface of a Gaussian mixture distribution. Is such an iso-density surface a union of ellispoids? Let's say that this Gaussian mixture ...
1
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1answer
71 views

What does rotational invariance mean in statistics?

What does rotational invariance mean in statistics? The property that the normal distribution satisfies for independent normal distributed $X_i$, $\Sigma_i X_i$ is also normal with variance $\Sigma_i ...
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1answer
26 views

Statistics questions (normal distribution and possibly gamma function)

This is a question from a past stat exam that I am studying because my final is in two days (lol). It'd be great if someone could guide me through how do both parts of the problem. I know gamma ...
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2answers
87 views

Adding and Subtracting Normal Distributions

I think I know how to do this, but I'm not sure. I'm just hoping to check myself here before I do a bunch of work incorrectly. Suppose you have three independent normal distributions: Distribution A: ...
2
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1answer
59 views

Probability of a point from one normal distribution being higher than a point from another independent normal distribution

Given two independent normal distributions: Distribution 1: Mean $= 23.95$, SD $= 7.44$ Distribution 2: Mean $= 16.29$, SD $= 7.79$ How often on average will a point from Distribution 2 be greater ...
3
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1answer
144 views

Calculating the probability of winning roulette after x bets

I'm going through all of my homeworks to study for my final and I'm getting hung up on this one problem I never figured out... A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you ...
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1answer
70 views

When using the Central Limit Theorem, how to scale the mean and variance depending on the number of samples?

So I'm reviewing my notes for the central limit theorem for my final and I'm getting hung up on one detail. The two questions below both utilize the central limit theorem, but they use it in ...
2
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1answer
31 views

Find the unit vector so that this condition is true.

Let $(X_1,X_2)$ be jointly normal with density $$\phi(x_1,x_2;\rho) = \frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left(\frac{-1}{2\sqrt{1-\rho^2}}(x_1^2 - 2\rho x_1x_2 + x_2^2)\right)$$ Find unit vector ...
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2answers
50 views

How to show that normal random variables U1 and U2 are independent?

Prompt: Assume that $Y_1, Y_2, Y_3$ and $Y_4$ are independently and identically distributed $N(\mu,\sigma^2)$ random variables. Show that $Y_1 + Y_2 – Y_3 – Y_4$ and $Y_1 – Y_2 + Y_3 – Y_4$ are ...
3
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1answer
46 views

Conditional probability for two normal distributed variables.

I haven't had to do much with probabilities since university, so please excuse if this is trivial or the question is not well specified. Let $X$ and $Y$ be two independent, normally distributed ...
1
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1answer
29 views

Normal approximation of Poisson Distribution

Hi currently studying for a final exam and I just want to confirm my approach/answers to this problem are correct: Suppose that $X \sim \mathrm{Poisson}$. We wish to test $H_0: \lambda = 50$ vs ...
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1answer
22 views

Is there a rule that can be used to easily approximate the pdf(x) for normal distribution?

Given the Normal Distribution with mean Mu and variance Sigma. With the respect to the rule of 3 Sigma, can one use similar estimations for the value of probability density function within 1, 2, ... ...
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1answer
94 views

Word problem related to normal distribution

Consider the following problem: WINK, Inc. is made up of 450 employees who work a total of 13,500 hours per week. If the number of weekly work hours per person has a normal ...
0
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1answer
97 views

Linear transformation of random variables

We have to stochastic variables X and Y, and we define $ \begin{pmatrix} \tilde{X} \\ \tilde{Y} \end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X \\ Y ...
1
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1answer
471 views

Expected value of lognormal distribution.

Hi I'm stuck on this question: Recall that X is said to have a lognormal distribution with parameters $\mu$ and $\sigma^2$ if log(X) is normal with mean $\mu$ and variance $\sigma^2$. Suppose X is ...
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0answers
25 views

normal approximation to poisson

Cotton yarn is wound onto bobbins, each of which takes $100$m of yarn. If the thread breaks before $100$m is reached, the bobbin is rejected. In a trial of a new spinning machine, $13$ bobbins out of ...
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2answers
78 views

The product of a normal and Bernoulli variables, independent from each other

Let $X\sim N(0,1)$ and let $Z$ be a random variable independent of $X$ such that: \begin{equation} \Pr(Z=z) = \begin{cases} \frac{1}{2} & \mbox{if $z = -1$ or $z=1$}, \\ \\ 0 & ...
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34 views

Why we can use normal distribution to approximate binomial distribution when n is large enough?

Prove why we can use normal distribution to approximate binomial distribution when n is large enough. Hint: Try to read something on bernoull ...
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0answers
27 views

questions about distribution of multivariate normal

I'm looking at this past exam question, For A) Cbhat~N(CU,C(summation)C') B)I have very faint idea of what to do, I tried finding some theroems about distribution but couldn't find any that ...
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2answers
99 views

Evaluating Integral involving exp

I am stuck at the following integral :- $$ \int_{- \infty }^{ \infty } {1\over x}\exp\left(-x^2-\frac{1}{x^2}\right)\,dx$$ Can anybody give me some hint. and also for this function $$ \int_{- ...
2
votes
1answer
22 views

Probability of an event occurring more than X% of the time

Cans are a normal random variable with a mean of 7.96 ounces and a standard deviation of 0.22 ounces. Suppose that you draw a random sample of 34 cans. Use normal approximation to find the ...
0
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1answer
30 views

Transforming a normal distribution to a uniform one

I'm searching for a method that transforms a normal distribution into a normal distribution. I've looked everywhere, but I'm not sure if I just missed something completely obvious, if this actually is ...
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0answers
22 views

How can I calculate $\int^{\infty}_{-\infty}\Phi\left(\frac{w-a}{b}\right) \left(\frac{w-a}{b}\right) f(w; \mu, \sigma²)\,\mathrm dw$

Suppose $\Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution and $f(\cdot; \mu, \sigma²)$ is the density of the normal distribution with mean $\mu$ and standard ...
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2answers
39 views

Product of a Continuous and Discrete Distribution

Let $X \sim \mathcal{N}(0, 1)$ and $Y$ be a random variable independent of $X$ such that \begin{align*} P(Y=y) = \begin{cases} \frac{1}{2} & y = -1\\ \frac{1}{2} & y = 1\\ 0 & ...
0
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1answer
41 views

Limit distribution of $X_n+Y_n$ if $X_n \overset{d}{\longrightarrow}X$, $Y_n \overset{d}{\longrightarrow}Y$ if $(X,Y) \sim N(\mu,\Sigma)$?

Let $X_n, Y_n$ be sequences of RV with $X_n \overset{d}{\longrightarrow} X$ and $Y_n \overset{d}{\longrightarrow} Y$ so that $\begin{pmatrix} X\\ Y \end{pmatrix} \sim N\left(\begin{pmatrix} ...
0
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1answer
43 views

Integrating the normal distribution over rational numbers?

Is it possible to integrate the normal distribution over rational numbers? What is the value of such integral? Is it $\pi$ minus the integral over irrational numbers?
1
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1answer
38 views

Mixed Conditioning - Two Normal Distributions

Let $Z \sim \mathcal{N}(0,1)$ and $Y|Z \sim \mathcal{N}(Z, 1)$. Show that $f_{Z|Y}(z|y)$ is a normal density, and find the parameters of this density. What I have so far: \begin{align*} ...
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0answers
19 views

Conditional expectations of joint normal distribution

$u_1$ and $u_2$ are jointly normal, with zero means, unit variances, covariance $\sigma _{12}$. I know $E(u_1|u_2)=\sigma _{12}u_2$, but why $E(u_1|u_2<c)= \sigma _{12}E(u_2|u_2<c)$ ?
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1answer
21 views

Identity involving the relation Normal Distribution and Other arbritary Distribution

Let $X$ be a continuous random variable taking value at $\mathbb R$ with distribution $F_\theta$ and density $f_\theta.$ Define a function $\psi_\theta:\mathbb R\mapsto \mathbb R$ by $$ ...
2
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2answers
36 views

Statistics: Odd Moments

Need help with this stat question. I know you start by integrating $z^k f(z)$ from $-\infty$ to $0$ + integral of $z^k f(z)$ from $0$ to $\infty$. After that I'm stuck.
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1answer
37 views

Expectation of random varible with normal distribution composed with exponential [duplicate]

I am trying to find $\mathbb{E}(e^{-X})$ where $X$ is a random variable with a general normal distribution. I end up with $$(2\pi \sigma)^{-\frac{1}{2}} \int_{-\infty}^{\infty} ...