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

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15 views

Range of sum of Normal Distribution.

May be its silly question but I was just wondering is there any way to find out the absolute range of sum of values of Random normal distribution of N numbers with mu and sigma as mean and Std. Dev. ? ...
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0answers
6 views

Certain parasite Trypanosome

Consider the length of individual trypanosome chosen at random from the population. Find a) Pr{20 < length < 30} I just say 0.41 + 0.21 = 0.62 b) Pr{length ˃ 20} 0.41 + 0.21 + 0.03 = 0.65 ...
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1answer
17 views

Integral on complex plane of a gaussian times power

I can't solve the integral $$ I = \int_\mathbb{R} \int_\mathbb{R} \ (x + i y)^{2k} \ e^{\displaystyle - \frac{(x + i y)^2 R^2}{1+R^2} - y ^2} d x d y $$ which can be rewritten as $$ I= \int_\mathbb{R} ...
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1answer
20 views

If X ~ N(0,σ^2), find the density of Y = |x|

If X ~ N(0,σ^2), find the density of Y = |x| Hi I am reviewing for an upcoming exam, and came across this question in the textbook. Can someone please help me with this question. Thanks
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1answer
15 views

Normal distribution of juice

Quantity of juice in a pack of 1L is normaly distributed with average (mean) 950ml, and with standard deviation of 10ml. What is the probability that random pack of juice contains less then 945ml of ...
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1answer
22 views

Intuition behind Normal distribution forumula

In this formula $$ P(x) = \frac{1}{{\sigma \sqrt {2\pi } }}e^{ - \frac{ \left( {x - \mu } \right)^2 }{2\sigma^2}} $$ why do we divide by square root 2 pi and after that multiply everything by e in ...
3
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1answer
44 views

$P(X^2+Y^2<1)$ of two independent n(0,1) random variables

Suppose that X and Y are independent n(0,1) random variables. a) Find $P(X^2+Y^2<1)$ Attempt: a) Let $U = X^2 + Y^2$, $V = Y$. Then $X = \sqrt{V^2 -U}$, $Y = V$. $J = \left| ...
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1answer
11 views

Percentages in Normal Distribution

A statistics problem involves: Lengths of a certain type of carrot have a normal distribution with mean 14.2 cm and standard deviation 3.6 cm. (i) 8% of carrots are shorter than c cm. Find the value ...
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1answer
37 views

Do we need to use continuity correction if we use CLT to do normal approximation

In a hotel, large number of cups and saucers are washed everyday. The number of cups that are broken each day while washing averages $2.1$. The number of saucers broken each day averages $1.6$, ...
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0answers
19 views

IQR of the sampling distribution of S^2

Suppose $X_1,X_2,...,X_5$ is a random sample from a $N(33,9)$ population and let $S^2$ be the sample variance. [a] Find the IQR of the sampling distribution of $S^2$. [b] Find the IQR of the ...
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0answers
8 views

Normal distribution around and extreme value

I'm trying to create an artificial dataset with users, items, and ratings given by the users to the items. Creating the dataset, I pick the average rating for every item randomly, and let the ratings ...
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1answer
27 views

Standard Normal Distribution and CDF

I have a data set which consists of measured time in seconds. Secs= ${3000, 3857, 2400, 3323}.$ Mean $\mu =3145$. Standard deviation $\sigma=609.556$. I calculated the Standard Normal variable for ...
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0answers
35 views

Normal distrubition [on hold]

Let Xi denote the weight of a randomly selected prepackaged one-kilogram bag of potatoes. Of course, one-kilogram bags of potatoes won’t weigh exactly one kilogram. Actually, history suggests that Xi ...
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0answers
69 views
+100

Existence of a bounded function satisfying a second order differential equation

This question is a variation version from here. Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ ...
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0answers
14 views

What is the normal distribution?

1) A researcher is interested in estimating the proportion of families with children in a certain district. To produce a 95% confidence interval of width less than 0.1, what is the minimum safe (or ...
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0answers
18 views

What type of distribution can be used to describe a game with positive expected winnings?

I've come across something I'm not too sure about. Let's say we flip a coin, heads mean we lose 1 unit, tails means we win a 1 unit. This distribution of outcomes in this would be considered normal, ...
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0answers
9 views

finding sampling distribution from standard normal

Let $X_1$, $X_2$, $X_3$ be a random sample of size $3$ from a standard normal distribution. Find the distribution of $X_1^2 + X_2^2 + X_3^2$. If possible, find the sampling distribution of ...
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0answers
11 views

completely lost. sampling distribution associated with normal populations. [closed]

Let X1,X2, X3, X4 be a random sample of size 4 from a standard normal population. Find the sampling distribution (if possible) and moment generating function of the statistic 2X1^2+3X2^2+X3^2+4X4^4. ...
2
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1answer
46 views

Boundedness of an integral of square function implying zero integral

Let $\alpha:\mathbb R\mapsto\mathbb R$ be the smooth function such that $$\int_{-\infty}^{\infty}[\alpha'(x)-x\alpha(x)]^2e^{-\frac{x^2}2}dx<\infty.$$ I wish to prove that ...
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0answers
19 views

Log-likelihood of the normal distribution.

On the attached picture I've highlighted the term which I do not agree with. Is it actually true ? In my calculations I get $$-n(\frac{1}{2}\log(\sqrt{2\pi})+\log(\sigma)),$$ instead. Thank you in ...
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1answer
32 views

Beginner Econometrics question about probabilities for a normal variable

$Y \sim N(\mu, \sigma^2)\implies (Y-\mu)/\sigma$ Prove that this has a Mean of $0$ and a Variance of $1$.
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1answer
13 views

Testing statistic $\frac{MSS(X)}{MSS(Y)}$

Suppose a test statistic $\frac{MSS(X)}{MSS(Y)}$, where $MSS$ denotes Mean Sum of Squares, is to be used for testing the significance of the factor $X$. Do we need the assumption $$\mathbb ...
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2answers
43 views

Proof that if $Z$ is standard normal, then Z^2 is distributed Chi-Square (1).

Suppose that $Z\sim N(0,1)$ and let $V=Z^2$. Prove that $V\sim \chi^2(1)$. I want to use the method of moment generating functions, because I already understand the proof using the method of ...
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0answers
42 views

Poisson process. Finding 5th and 95th centiles

I am an undergraduate student of Economics. Today I was trying to solve 1 exercise related to Poisson process that I found confusing and I would be very grateful for your help, as my Mathematics ...
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1answer
63 views

Computing the characteristic function of a normal random vector

The characteristic function of a random vector $\boldsymbol{X}$ is $\varphi_{\boldsymbol{X}}(\boldsymbol{t}) =E[e^{i\boldsymbol{t}'\boldsymbol{X}}] $ Now suppose that $\boldsymbol{X} \in ...
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1answer
44 views

Finding the 99% of a normally distributed graph

The heights of adults are normally distributed with a mean of 187.5 cm and a standard deviation of 9.5 cm. A standard doorway is designed so that 99% of adults have a space of at least 17 cm over ...
2
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0answers
38 views

Finite discrete approximation to the normal distribution

I wish to derive a finite (that is, which has a finite support) discrete approximation to a normal distribution, with the following considerations: It should have exactly the same mean and variance ...
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0answers
21 views

Can anyone help with this discrete problem? [duplicate]

Question: How many ways are there to distribute 16 identical pieces of candy to five children such that every child receives at least one piece? Generalize k identical pieces of candy and n children. ...
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2answers
49 views

What is the reason for this answer on this coin problem?

Question: How many ways are there to pick a collection of 15 coins from bags of pennies, nickels, dimes, and quarters? (Assume coins of the same denomination are indistinguishable.) I know the answer ...
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1answer
36 views

Variance of the increment in Brownian Motion

How do you show that variance of $B_t-B_s$ is $ t-s$. where $B_t$ = $\int p(t,x,y)dy$ where $p(t,x,y)$ = $ (1/ \surd(2\pi t))\exp(-(x-y)^2/2t$ where t>0 and $x,y \in \mathbb{R}$
1
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1answer
37 views

Calculation of the $n$-th moment of the normal distribution

I need some advice on a question, maybe someone can give me a hint. Let $X \in N(0,\sigma^2)$, show that $E[X^{2n+1}] = 0$ for $n = 0,1,2,\ldots$, and that $E[X^{2n}]= [(2n)!/2^nn!]\cdot ...
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1answer
31 views

Stuck during calculations to show that $\mathbb{E}e^{uX}=e^{u\mu + 0.5u^2\sigma^2}$ for normal random variable $X$

I am reading S. Shreve's introduction to Stochastic Calculus. Exercise 1.6. (p. 43) should be a simple exercise, but I don't know how to continue. Let $u$ be a fixed number in $\mathbb{R}$ and $X$ a ...
2
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2answers
81 views

Expected distance between two vectors that belong to two different Gaussian distributions

Let $X$, $Y$ be two random variables that follow the Gaussian distributions with mean vectors $\mu_x$, $\mu_y$, and covariance matrices $\Sigma_x$, $\Sigma_y$, respectively. The probability density ...
1
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4answers
33 views

when to use which z score equation?

in some exam past papers I have been doing I have come across the statistics equation z=(sample mean - mean)/standard deviation as well as the equation z = (sample mean - mean)/(standard ...
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0answers
34 views

Ito formula for jump proccess

I have just learned Ito fomula for jump processes but I have still not understood it well. Assume that I have $dS_r=S_{t^-}\mu+S_{t^-}\sigma dB_t +S_{t^-}\int_{\mathbb{R}^+}(y-1)N(dt,dy), \;\; 0\leq ...
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2answers
148 views

Normal distribution in which 90% of samples are between 2.99 and 3.01; what is the standard deviation?

Steel rods are manufactured to be 3 inches in diameter but they are acceptable if they are inside the limit 2.99 inches and 3.01 inches. It is observed that 5% are rejected as oversized and 5% are ...
2
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2answers
69 views

Find distribution $Y=X^2$

X~N(0,1). Find distribution $Y=X^2$ Can someone help me? I have no idea how to do it. I could try to start like this: $F_Y(t)=P(X^2<t)=P(-\sqrt(t)<X<\sqrt{t})$
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0answers
30 views

Maximum of a Gaussian random walk with non-identical steps

Consider a sequence of independent normal random variable $X_1,...,X_n$ with (negative) means $\mu_1,...,\mu_n$ and standard deviation $\sigma_1,...,\sigma_n$. Define \begin{equation} S_k = ...
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2answers
31 views

Errors and Residual

Why are errors independent but residuals dependent? As far i know the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. But also ...
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1answer
38 views

Normal Distribution Worded Problem

Standard deviation = 2.5 mL 98% of bottles must be between 998 mL and 1000mL Pr( 998 < x < 1000) = 0.98 This is a technology exam question, therefore to find the mean I used the method: ...
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0answers
16 views

Generate Correlated Normal and Log-Normal Random Variable

The standard approach for generating two normally distributed random variables some with correlation $\rho$ is explained here: Generate Correlated Normal Random Variables. Now let $X,Y$ be normally ...
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0answers
31 views

Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution

As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there ...
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1answer
20 views

Standard deviation in normal distribution

A manufacturer uses a machine to make metal rods.The diameter of the rods follow a normal distribution with a mean of 1cm and a standard deviation of 0.02cm If the standard deviation of the diameters ...
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1answer
40 views

Convergence sequence of random variables

I have this problem about a sequence of normals. $(X_n)_{n\geq 0}$ is defined as $$X_{n+1}=aX_n+U_{n+1}$$ $X_0=0$, where $(U_n)_{n\geq1}$ is a sequence of i.i.d random variable normally distributed ...
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0answers
6 views

References to papers/books that uses a kernel to smooth a discrete distribution

Since a kernel, such as Gaussian, is often used to smooth out the distribution of discrete points in 1D, 2D or 3D, I believe there must be some study materials or research work that have used this, ...
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3answers
51 views

Transformation(?) of Random Variables

There are two independent Gaussian R.Vs: $U:N(-1,1)$ and $V:N(1,1)$ How do I go about finding the PDF of the following transformations? X = U+V T = (U+2V, U-2V) W = U (with 50% chance), V (with ...
2
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1answer
13 views

Distribution of a function of a normally distributed variable

Let's say you have a random variable $X$, which is normally distributed according to $X \sim \mathcal{N}(1,2)$. With $1$ being the mean and $2$ being the variance. Now let's say that there is another ...
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1answer
36 views

How to apply Gaussian kernel to smooth density of points on 2D (algorithmically)

I have a set of points on a 2D surface and need to build a heatmap. However, I also need to smooth out the density/distribution by applying some sort of kernel (Gaussian kernel, for example). I Know ...
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0answers
21 views

How to apply Gaussian kernel to smooth density of points on 2D (algorithmically) [duplicate]

I have a set of points on a 2D surface and need to build a heatmap. However, I also need to smooth out the density/distribution by applying some sort of kernel (Gaussian kernel, for example). I Know ...
0
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1answer
20 views

Expectation of an exponentiated quadratic form

Given a multivariate normal random $n\times 1$ vector $X \sim N(\mu,\Sigma)$, what is the expectation $$\mathbb{E}[exp(X^TAX+b^TX)]$$ where $A$ is a $n\times n$ matrix and $b$ is a n-dimensional ...