Tagged Questions

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

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0
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0answers
12 views

Marginalizing product of multivariate normal distributions

How should I marginalize $F_{i}$ from the following probability distribution $$p(y_{i}|F_{i},\alpha, \Lambda, \Phi, \Sigma) = N(\alpha + \Lambda F_{i}, \Sigma)$$ in order to obtain ...
-3
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0answers
20 views

Normal Distribution(Probability) [on hold]

(a) The hourly wage of employees in a certain service industry is believed to follow the Normal Distribution N(40, 5^2) which has a mean µ of 40 dollars and a standard deviation σ of 5 dollars. The ...
-2
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0answers
16 views

Poisson and Normal Distribution [on hold]

Hits to a high-volume Web site are assumed to follow a Poisson distribution with a mean of 10, 000 per day. (a) Approximate (by a normal distribution) the expected number of days in a year (365 days) ...
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0answers
13 views

How to check null hypothesis in Minitab without specific data?

I have been given a question that specifies a sample size of 50, sample mean of 3.05, standard deviation of .34, and desired mean of 3.2. The question asks whether or not the average mean is ...
0
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1answer
6 views

relationship between two normally distributed variables

Say I have two normally distributed independent random variables (X1 and X2) with the same variance but different means. How would I calculate P(X1 > X2)?
0
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0answers
13 views

Multivariate Normal Variable (Homework)

I am trying to solve the following question: Let $(V, Z) ∼ MVN(0,I)$ (where I is the identity matrix) and let $Y=V+Z+1$. Find the distribution of $(1+Z,1-Y)$. I have found the distribution of ...
1
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0answers
17 views

Normal pdf/cdf inequality

Let $\Phi$ be the cdf and $\phi$ the pdf of the standard normal distribution. I want to show that: $$ \Phi(z)[1-\Phi(z)]\geq \phi(z)^2, \quad z\in\mathbb R. $$ How can I do this? I tried by looking at ...
0
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2answers
7 views

normal distribution question with percentages

how a can i solve a normal distribution without the mean ? suppose a truck of river sand delivered by a company has normal distribution with a standard deviationof 100kg.if 20% of loads are at least ...
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1answer
24 views

Distribution for random variable Z = Y1 - Y2

This was one of the interview questions. I did not know the answer. Question : Let Y1 and Y2 be two independent random variables where Y1 follows Normalpdf[x, -2, 5] distribution and Y2 follows ...
1
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0answers
25 views

What is $E[\cos X]$ where $X$ is lognormal?

I was asked in an interview to compute $E[\cos X]$ where $X$ is lognormal. I tried using lognormal's characteristic function (Taylor series representation, which is divergent) and $\cos ...
2
votes
0answers
28 views

How to calculate this kind of probability for a normal distribution?

here is my question. I have a normal distribution with known mean and variance. Say the mean is 3 and the Var. is 2. what is the probability that the random variable is taking value 2.9? If I plug ...
1
vote
1answer
27 views

When to expect normal distribution?

I was wondering when a normal distribution can be expected. I know that things like: heights of people size of things produced by machines errors in measurements blood pressure marks on a test ...
1
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2answers
18 views

Studies shown that gasoline use for compact cars sold in the U.S. is normally distributed, with a mean of 25.5 mpg and standard deviation of 4.5 mpg.

Studies shown that gasoline use for compact cars sold in the U.S. is normally distributed, with a mean of 25.5 mpg and standard deviation of 4.5 mpg. Find the range of mileage for the middle 60% of ...
0
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0answers
21 views

How to generate random normal skewed distribution?

We have right skewed normal distribution dataset whose mean is ~180, SD is ~60 and Skewness is 1.64. We have calculated Skewness using skewness function of R package "e1071" How do we generate ...
0
votes
0answers
13 views

Math Statitics. Normal Distrubuton [on hold]

Studies shown that gasoline use or compact cars sold in the U.S. is normally distributed, with a mean of 25.5 mpg and standard deviation of 4.5 mpg. Find the range of mileage for the middle 60% of ...
3
votes
1answer
30 views

Find the cutoff level for the highest 15% in normal distribution, given the mean and standard deviation [on hold]

The cholesterol levels of adult American women are approximately normal with the mean of 188 mg/dl and a standard deviation of 24 mg/dl. a company wants to test a certain medication for women ...
-1
votes
1answer
36 views

Exponential to Normal approximation [on hold]

Suppose that the length of life of a piece of equipment is exponentially distributed with a mean length of 30 days. As soon as it fails another is installed in its place. Find the probability that ...
0
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0answers
25 views

Airplane Overbooking Problem

Sometimes customers will make a reservation and then not turn up. To off-set this problem some companies may decide to “overbook” so they are not left with empty places. For example, an airline ...
1
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1answer
21 views

Applying a Normal Distribution to Another Function to Find Probability

Suppose that the number of hours students spend studying for an exam is approximately normally distributed with $\mu=10$ and $\sigma=\sqrt{2}$. If a student spends $t$ hours studying, he/she ...
0
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0answers
10 views

Suppose the number of hours that a student spends working on an assignment is approximately normally distributed…

with mean $\mu = 10$ and variance = 2. If a student spends t hours working on the assignment she receives a mark of M(t): $M(t) = \frac1{1 + e^{-t+7} }$ What is the probability she receives at least ...
1
vote
1answer
32 views

Box-Muller Transformation

I know that we can use the Box-Muller transformation to generate a pair of independent standard Gaussian random variables using a pair of independent standard uniform random variables. I am wondering ...
1
vote
1answer
7 views

How can I find the percentile function of a distribution that isn't normal?

I know that: $$ X = \mu + Z\sigma$$ for a normal distribution. I'm having a tough time understanding where this is derived from, though. How is it found and how is it found for other distributions?
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0answers
18 views

Transformations of Normal Distribution

Let $X \sim \mathcal{N}(0, 1)$. We define the CDF, $\Phi(x)$, of $X$ as: $$ \Phi(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} e^{-\frac{t^2}{2}}\,\mathrm{d}t $$ If $Y=\Phi(X)$, what is ...
2
votes
2answers
51 views

You purchased stock for \$1m. What is the probability that it is worth more than $30m after 10 years?

The change in value of the investment each year is modeled as follows: Divided by 2: 1/4 Remain unchanged: 3/8 Doubles: 1/4 Quadruples: 1/8 Where I'm at: I'm aware that this needs to be formulated ...
1
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2answers
51 views

Let X be normally distributed with mean $0$ and variance $1$, find the CDF and density of $Y = \Phi(X)$

Define $\Phi(x)$ as: $$ \Phi(x) = \frac 1{\sqrt{2\pi}}\int_{-\infty}^x \exp\left(-\frac{t^2}{2}\right) dt $$ and let the random variable $Y$ be defined as $\Phi(X)$ where $X$ is a standard ...
0
votes
1answer
9 views

Converting normalised values into original

I have a normalisation formula as follows, which takes a list of numbers, such as $1,2,3,4,5,6,7,8,9,10$, and returns the normalized values which guarantees that $\tilde{x_i} \in [0,1]$. ...
1
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0answers
20 views

Sampling Distribution of the Mean

I want to know if my reasoning is correct. Let's say I got two normal distributed variables: Variable "X": 5.4 (mean), 2.856 (variance) Variable "Y": 5.4 (mean), 5.062 (variance) Let's pick 16 ...
1
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1answer
26 views

How do I plot normal distribution

If I know the range (1-24) and know the area (X), how can I plot a normal distribution so that the curve has area X?
5
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1answer
34 views

Total variation distance of two normal random variables $X_t \sim \mathcal{N}(0,s)$ and $X_s \sim \mathcal{N}(0,t)$

I need to prove that the total variation distance between two normal random variables $X_t \sim \mathcal{N}(0,s)$ and $X_s \sim \mathcal{N}(0,t)$ converges to $0$ when $s \nearrow t$. We know that ...
-1
votes
1answer
15 views

How to find probability $P(x_1<x<x_2)$ from sampled data [closed]

Supposed $n$ samples exist, and we know that $P$ is normally distributed with $\mu$ as mean $\sigma$ as standard distribution. What will be $P(x_1 <x <x_2)$ inferring approximately from sample ...
-1
votes
0answers
9 views

How to estimate probability P(x_1 < x < x_2) when $P$ is normally distributed using Chi-squared distribution and sampled data [closed]

Suppose that $\mu$ and $\sigma$ is unknown. Sampled data from finite $n$ data show that $\bar{X}$ as sample mean and $S^2$ as unbiased sample variance. Using this information, how does one use ...
0
votes
1answer
47 views

Estimating the probability of a failure

We are estimating the spares requirement for a radar power supply. The power supply was designed with a mean ($\mu$) life of $6500$ hours. The standard deviation ($\sigma$) determined from testing is ...
1
vote
1answer
42 views

Prove that $Y = \frac{X_1+X_2*X_3}{\sqrt{1+X_1^2}}$ obeys normal distribution

given that $X_1, X_2, X_3$ are independent and identically distributed, $X_1 \sim N(0,1)$. I tried to calculate the cumulative distribution function of Y: \begin{align} P(Y\leq y) &= ...
0
votes
1answer
12 views

sum of iid variables: how many terms needed for convergent to normal

For sum of iid variables $Z_n=\sum_{i=1}^nX_i$, in general, how large should $n$ be to indicate 'convergence' to normal? 10? 100?
0
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1answer
27 views

Correlation coefficient and Expectation of two dimensional normal distribution.

Random variable (X,Y) is normally distributed. Conditional expectations are $E(X|Y=y)=0.25y + 2$ $E(Y|X=x)=x-2$ How can i determine correlation coefficient and when that is known, the expectations ...
0
votes
1answer
29 views

Approximation of uniform distribution.

There are leaving from the station arriving every 10 minutes. A person has to wait from 0 to 10 minutes at the station, this is uniformly distributed. Now if the person uses the station 100 times a ...
1
vote
1answer
30 views

finding the pdf for $y$

Let $X\sim N(0,1)$ that is $X$ is a random variable with normal distribution with mean$=0$ and standard deviation$=1$ and $$f_X(x)=\dfrac{1}{\sqrt{2\pi}}e^{\frac{-x^2}{2}}$$ Let $y=g(X)=\dfrac{1}{x}$. ...
0
votes
0answers
13 views

Help relating gaussian to chi-squared distribution

I am having trouble finding a simple layout/documentation for the chi squared distribution. From what I understand the chi squared distribution is just: Where "v" is some strength parameter. Now, ...
0
votes
0answers
9 views

Computing CDF of Normal-Bernoulli

I have the following setting: Let $\theta \sim N(\mu,\sigma^2)$ and \begin{equation} e = \left\{ \begin{array}{l l} E & \quad \text{if} \quad \theta > \theta^* \\ 1 & \quad \text{if} ...
1
vote
3answers
18 views

Characteristic function of a square of normally distributed RV

Let's assume that $ X \sim \mathcal{N}(0,1) $. I'm supposed to compute the characteristic function of $ X^2 $. As far as I got is that the density of $ X^2 $ is $ g(y) = \frac{1}{2\sqrt{2\pi}} ...
1
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1answer
26 views

Characteristic function of a product of random variables

I am facing the following problem. Let $X,Y$ be independent random variables with standard normal distribution. Find the characteristic function of a variable $ XY $. I have found some information, ...
0
votes
1answer
45 views

Multiple hypothesis testing

Let's suppose I have 10 independent measurements with results close to zero. How can I claim that they are in agreement with the theory, them being zero? The errors of these 10 results are not equal, ...
0
votes
1answer
12 views

Is the linear transform of joint gaussian necessary gaussian? See this case!

Suppose we map the low dimensional Gaussian distribution into higher dimension using linear transform. Say, $X \in R^p$ is joint Gaussian, and for $n > p$, $Y = A_{n \times p}X$. Is $Y$ joint ...
0
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0answers
83 views

Using the central limit theorem to prove a statement regarding normal distribution, from a population with exponential distribution

X1, . . . , Xn are a random sample from a population having an exponential distribution with rate parameter λ. Use the Central Limit Theorem to show that, for large values of n, sqrt(n)*(λx − 1) ∼ ...
1
vote
3answers
65 views

Probability of two normal random variables when random samples are taken from a population

This is sort of second section to my previous question, I should have included both together, but I forgot to. Sorry for any inconvenience. X= random height of a male Y= random height of a female X ...
1
vote
2answers
117 views

Probability of a normal random variable added to a number being greater than another normal random variable, and distribution of average

$X$= random height of a male $Y$= random height of a female $X$ and $Y$ are independent of each other For $x$, $\mu=180\text{ cm}$ and $\sigma^2= 16\text{ cm}^2$ For $y$, $\mu=170\text{ cm}$ and ...
1
vote
1answer
8 views

The effect of a decrease in sample size on a confidence interval

I have a some data that contains 100 elements. I can model it as a normal distribution using the t-distributionI have used the t-distribution to construct a confidence interval for unknown value of ...
1
vote
0answers
14 views

If Gaussian random vector has singular covariance matrix, isn't there probability density function?

I got a complicated problem. Suppose that Gaussian random vector having Covariance matrix; $$K_X=\left[\begin{matrix}1 &-\frac12 &-\frac12 \\ -\frac12 &1 &-\frac12 \\ -\frac12 ...
1
vote
1answer
29 views

Determine probability of fewer than a certain number of events

Could anyone help with the following problem? My guts is telling me that the answer to part (a) is a normal distribution. Mainly, because I can't see where a uniform distribution would fit in this ...
1
vote
1answer
16 views

Calculating probability using normal tables

I've had a crack at this question however I don't seem to be getting the correct answer and I can't figure out why. I've been given a table of the 'Normal Distribution Function' where the left tail is ...