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

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Integral involving CDF of a normal distribution

Can we evaluate the following integral ? $$\int_0^\infty x e^{-x^2} \Phi(ax+b)\,\mathrm dx$$ Here $\Phi(\cdot)$ is the cumulative probability distribution function of a standard normal random ...
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53 views

Convolutions and the Gaussian distribution

Suppose $X_1$ and $X_2$ are independent random variables each with the standard Gaussian distribution. Compute, using convolutions, the density of the distribution of $X_1 + X_2$ and show $X_1 + X_2 = ...
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3answers
173 views

Normal distribution notation

I am wondering... is saying $\mathcal{N}\left(0,\begin{bmatrix} 0.1 & 0.02 \\ 0.02 & 0.3 \end{bmatrix}\right)$ equivalent to $\mathcal{N}\left(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{...
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65 views

How to calculate the Gaussian Integral in specific region?

Firstly, I know that the Gaussian Integral formula, e.g., $\int^{+\infty}_{-\infty}e^{-ax^2}dx=\sqrt{\frac{\pi}{a}}$. But, I am now being encountered a problem when the integral region is not $[-\...
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195 views

z score of normal distribution

Good day, I want to ask about standard normal distribution. What is the highest and lowest value of $z$ score can be? From the table of standard normal, the value $z$ score is only for -3.99 $\leq$ $...
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139 views

Sum of dependent normal random variables

Let ${\bf X} =(X_1,\ldots,X_n)'$ be a vector of random variables that may be dependent and let ${\bf a}=(a_1,\ldots,a_n)'$ and ${\bf b}=(b_1,\ldots,b_n)'$ be nonrandom vectors with $a_i \neq 0$ and $...
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341 views

Just learned about the bell curve in statistics. How is calculus related to this curve?

I'm learning about the bell curve in statistics and I'm trying to understand the calculus behind the concept. I've taken calc 1 already. How is the integral related to this ...
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277 views

Normal distribution without standard deviation given

The proportion of pink candies in a bag is supposed to be $50\%$. The filling machine is to be tested to see if it fills with the right proportion. A random sample of $50$ candies is made. The machine ...
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41 views

Probability distritubion of linear function

Given a variable X belongs to gaussian distribution $N(\mu, \sigma)$. How to find the distribution of linear function $y=ax+b$? My answer is that the linear distribtion will belong the $N(a\mu,\sigma)$...
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71 views

Forth Moment of Sum of Normal with Equal Correlation

I have $X_1,\dots,X_n$ identically normal distributed $N(0,\sigma^2)$ and $\operatorname{corr}(X_i,X_j)=\rho $ for all $i\neq j$. I'd like to compute \begin{equation} E\left(\sum_{i=1}^nX_i\right)^4. ...
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138 views

Bivariate distribution of the sum and product of Gaussian distributed numbers

If $X$ and $Y$ are independent normally distributed random variables $$X,Y\sim\mathcal{N}(0,\sigma^2)$$ How are the sum and product, $X+Y$ and $XY$, co-distributed? You can write the moment ...
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118 views

A sequence of Gaussian random vectors converges to a Gaussian random vector

Suppose $\left\{X_n : n \in N\right\}$ is a sequence of Gaussian random vectors and $\lim_n X_n = X$, almost surely. If $b := \lim_{n\rightarrow\infty} EX_n$ and $C := \lim_{n\rightarrow\infty} \...
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456 views

Differentiating mahalanobis distance

I would like to differentiate the mahalanobis distance: $$D(\textbf{x}, \boldsymbol \mu, \Sigma) = (\textbf{x}-\boldsymbol \mu)^T\Sigma^{-1}(\textbf{x}-\boldsymbol \mu)$$ where $\textbf{x} = (x_1, .....
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1answer
1k views

Characteristic function of Normal random variable squared

Probability density function of $X^2$ when $X$ has $N(0,1)$ distribution While reviewing above, Why do you sub $X^2$ for the $Y$ in $e^{tY}$ and not the density of the normal $(\frac{1}{\sqrt{2\pi}})...
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166 views

Sum of Gaussian Variables may not Gaussian

I am currently trying to understand the following three points which we discussed in lectures recently: We say that $X=(X_1,\ldots,X_d)$ is $d$-dimensional multivariate Gaussian distributed if $X\...
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152 views

How to use Chebyshev Inequality

Use Chebyshev Inequality to estimate the probability that in any one day of a business that earns a mean of 100 dollars a day with a standard deviation of 28.87 dollars, that business will make either ...
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137 views

Standard Normal Distribution to Normal Distribution

Given function f() which returns a random value from a standard normal distribution, is it possible to define a function g(mu, sigma) which returns a random value from a normal distribution with a ...
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1answer
44 views

Cumulative Distribution Function of $X = Y*I + Z*(1-I)$

I have a random variable $X = Y*I + Z*(1-I)$ where $Y~N(u, sigma^2)\,,\,\, Z~N(u, sigmac^2)$, and $I~B(1,1-p)$. What I can't seem to figure out is how to get a cumulative distribution function for $...
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565 views

Mean and Variance Convergence with r.v.

Let $(X_n)_{n\ge 1}$ be a sequence of random variables, with respective distributions being Gaussian, with respective mean $\mu_n \in \mathbb R$ and variance $\sigma_n^2 > 0$. Prove that if $X_n$ ...
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342 views

Normal Distribution Question

Could someone go over these calculations and tell me where I'm going wrong please. It's to do with normal distribution. The question: In a factory, the packets of sweets produced are supposed to ...
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855 views

Probability that one normal distribution is greater than the other when they are correlated i.e (Not Independent)

Would like to know how to approach this question: Probability that one normal distribution is greater than the other when they are correlated i.e (Not Independent). Seen the solution for the ...
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206 views

Independent, Normally Distributed R.V.

Working on this: A shot is fired at a circular target. The vertical and the horizontal coordinates of the point of impact (with the origin sitting at the target’s center) are independent and ...
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1answer
795 views

Conditional probability distribution with Gaussian noise

If I have a relationship as follows: $$Y = a X + G(0,\sigma^2),\text{ so }y = a X + \text{some Gaussian noise}.$$ The conditional probability distribution of $y$ given $x$, i.e. $P(y|x)$, is equal ...
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156 views

identically distributed question

$X$ and $Y$ are independent normal random variables and have the same moment-generating function and are thus identically distributed. Find the distribution of $Z$ where $Z=aX+bY$. I have the MGF ...
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323 views

Find standard deviation

The assignment is given: In a heat regulated room, we have two temperature limits $T_{\text{min}} = 16$ and $T_{\text{max}} = 24$. If the temperature is between $T_{\text{min}}$ and $T_{\text{max}}...
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126 views

Covariance of bi-variate normal distribution

$ \left( \begin{array}{c} X_1 \\ X_2 \end{array} \right) \sim N\left( \left( \begin{array}{c} 0 \\ 0 \end{array} \right) , \left( \begin{array}{cc} 1 & r \\ r & 1 \end{array} \...
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1k views

Characteristics of a Gaussian on a logarithmic scale?

I have data modelled to be Gaussian-distributed, and then displayed on a logarithmic scale and like to know some properties of the displayed data. Effektively, I ignore that a Gaussian $X$ (with ...
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1answer
33 views

Find exact difference between two values in Normal Distribution

If we have a normal distribution of N(10,2) and we are asked on what is the proportion of values betwen 7 and 8 we can calculate this by: ...
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44 views

Calculate $E(X^{2n})$ where $X$ is normal (0,1)

I need help proving the following: Let $X$ be normally distributed with parameters $\sigma=0$ and $\mu=1$. Let $n$ be a positive integer. Show that: $$E(X^{2n})=\frac{(2n)!}{2^nn!}=:(2n-1)!!$$ I've ...
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22 views

Derivation of a property of standard Wiener processes

I am reading A Standard Wiener Process and am struggling to piece together how they arrived at their conclusion. The major properties of any Wiener Process are: $W(t) = 0$ $W(t) - W(s) \sim N(0, t-...
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17 views

Link between conditional characteristic function and conditional density

Let $X$ and $Y$ be random variables (real-valued). I define $$E[e^{i\theta X}\mid\sigma(Y)] =: g(Y,\theta)$$ Suppose that $g(Y,\theta) = e^{i\theta Y}e^{-\frac{1}{2}\theta^2}$. Can I then say that ...
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14 views

Confidence Itervals; $Z_{\alpha}$ & $Z_{\alpha/2}$

I'm confused about what exactly $Z_{\alpha}$ is, does there exist a formula for it in terms of $\alpha$? IF so, is there also one for $Z_{\alpha/2}$?
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32 views

get data to draw a gauss curve

I would like to know how to get some data from a normal distribution to draw its gauss curve. I have the standard deviation, the average and the x, but I don't know how to get some points to draw the ...
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37 views

How to solve $P(X=a) = P(X=b)$

A random variable X is normally distributed with $\mu = 60$ and $\sigma$ = 3. What is the value of 2 numbers a,b so that $P(X=a) = P(X=b)$. The solution is $a = 60$ and $b = 65$. However, I do not ...
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59 views

Find $f$ and $g$ such that $\Phi(x)f(y)=g(x+y)$

Let $\Phi: \mathbb{R} \to [0,1]$ define the standard normal cdf function. I am trying to find some functions $f$ and $g$ -- if they exist -- such that $$\Phi(x)f(y)=g(x+y)$$ for all $(x,y)\in \mathbb{...
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Combining answers from a group, what proportion would I expect to reject the null hypothesis?

The scenario is that a group of $n$ people carry out this same experiment. Each person generates 11 random numbers based on the normal distribution $\mu=2.7$ and $\sigma=0.6$. Each person must then ...
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63 views

chi squared distribution of independent normal distributions that are not standard normal

I've been working on the following problem. I'm a bit confused about some of the specifics of how to arrive at the correct answer. I hope someone here could point me in the right direction: A dart ...
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28 views

Estimating parameters of a linear model

Suppose there is $n$ data points $(x_i,y_i)$ and $i=1,...,n$, sampled from a line in 2D modelled by $y = m_n x + b_n$ where $m_n \sim \mathcal{N}(0,\sigma^2_m)$ and $b_n \sim \mathcal{N}(0,\sigma^2_b)...
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56 views

From normal to normal standard distribution + a bonus Q

I am failing to see where the $\sigma$ is going in the below. Given that normal distribution's pdf is: $$p(x) = \frac{1}{\sigma \sqrt{2 \pi}}\exp \left( -\frac{(x-\mu)^2}{2 \sigma^2} \right)$$ and ...
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32 views

Is Z in standardized normal distribution always positive?

The question asks me to find the mean, given that: $\sigma$ is $0.8$ when not standardized 96% is over 40 I worked out that $Z = -1.75$ in this case, which then would lead me to $$Z = (\mu-40)/\...
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85 views

Difference between the Error function and Normal distribution function?

I have just started reading about the Error function but Distribution theory is not my strong point. So I apologize in advance for asking really simple questions about it. So the Error function is ...
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47 views

The ratio of WHAT type of independent random variables is normal?

I know that the ratio of two independent normal random variables is a Cauchy random variable. The ratio of WHAT type of independent random variables is normal?
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118 views

Can I plot a normal probability distribution given the number of trials, average, minimum, maximum, and standard deviation?

I have the information about processing timing of db transactions. I was wondering if the info I have is sufficient to plot a normal probability distribution graph. Data available: 560000 hits ...
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48 views

Using Normal Distributions to find Proportion

The height of a randomly selected woman from a population is normal with $\mu=165cm$ and $\sigma=7cm$. The heights f the men in this population are normal with $\mu=178cm$ and $\sigma = 8cm$. I am ...
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What's $r$ going to be when you get the summation of $36$ Geometric $X_i$'s

Let $X_1,X_2,\ldots,X_{36}$ be a random sample of size $n=36$ from the geometric distribution with the p.d.f: $$f(x) = \left(\frac{1}{4}\right)^{x-1} \left(\frac{3}{4}\right), x = 0,1,2,\ldots$$ Now ...
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62 views

Normal distribution with sample

I'm trying to figure out the best approach to this problem. I would assume that I can use the Central Limit theorem first and then a binomial cdf: Chocolate is packaged into jars using a computerized ...
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93 views

How to I use the standard normal table to get the following Z value?

So I am given that $P(X \le 31.5) = 0.05$ and according to the textbook, after standardizing and using the standard normal table we get $$(31.5 - \text{mean})/(\text{standard deviation}) = -1.645.$$...
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126 views

Ratio of CDF to PDF increasing?

Let $\Phi(x)$ be a cumulative normal distribution function and $\phi(x)$ the associated probability density function. Is the ratio $\frac{\Phi(x)}{\phi(x)}$ increasing in x? Numerically it seems to ...
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23 views

Normal approximation to Binomial probability distribution

Where did this 0.5 come from? I understand we are using Z-score but in my calculations I basically omit the 0.5 to get a probability of .9616.
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48 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 ...