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

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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 ...
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163 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 ...
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151 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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136 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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10k views

Normal distribution with absolute value

I am new to the normal distribution topic. While I have understood and solved various different kind of questions, the normal distribution questions with absolute value, are the ones I have no idea ...
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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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543 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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338 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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848 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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204 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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788 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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155 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 ...
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309 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 ...
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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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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, ...
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15 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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58 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 ...
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17 views

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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61 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 ...
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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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30 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 = ...
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79 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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1answer
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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107 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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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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36 views

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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57 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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79 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}) = ...
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113 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 ...
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38 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 ...
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75 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 ...
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51 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 ...
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74 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 ...
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31 views

Formula to get mean and standard deviation of this multi-variable equation

$$ \binom n x \times\left(\frac1r\right)^x\times\left(\frac{r-1}r\right)^{n-x} $$ If you have $n$ boxes and have a $\frac1r$ chance to fill each one, this equation returns the chance that you fill ...
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91 views

Calculating probability of a normal distribution, not getting correct answer

I'm doing a homework assignment and having some trouble matching the correct answers from my professor. As a reference, I'm calculating these answers using R. The question is as follows: Assuming ...
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72 views

Integral of multplication of normal pdf and Rayleigh pdf distribution

I need to calculate following definite integral $$\frac{1}{2\pi }\int_0^\infty \frac{x^2 e^{-x^2/\sigma^2 } }{\sigma} \frac{e^{-\frac{\lambda}{{ax^2+b}}}}{\sqrt{ax^2+b}} ~~dx$$ It is actually ...
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70 views

Finding expected value of $|X_1+X_2|$, given that the two roots $X_1, X_2$ of $X^2 + 2BX + 1 =0 $ are real

$X^2 + 2BX + 1 =0 $ The random variable B is normally distrubuted with mean zero and unit variance. Given that the two roots $X_1, X_2$ are real, find, giving your answers to three s.f. the ...
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132 views

Gaussian distribution raised to a power

Given that $X$ follows a Gaussian distribution $e^{-x^2/2\sigma^2}$, what distribution is followed by $X^{1/3}$? How does one start to solve this problem? I guess it isn't ...
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37 views

Solving for an unknown $\mu$ in a probability problem involving normal random variables.

(a): $P[X < 355] = P[Z < \frac{355 - 360}{4}] = P[Z < -1.25] = 1 - \Phi[1.25] = .1056$. Part (a) is simple, but I included it because I was not sure if I should somehow use it to solve ...
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59 views

Determine $A$ such that $Q=X'AX$ has chi-squared distribution.

Let $\boldsymbol X\sim N_n(\boldsymbol\mu,\boldsymbol\Sigma)$, where $\boldsymbol\Sigma$ positive-definite. I am trying to determine, in general, what form $\boldsymbol A$ (one example is ...
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111 views

Proving completeness of the average of a random normal sample

Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show ...
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190 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 ...