What is Normal distribution: Definition and 240 Discussions

In probability theory, a normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is













{\displaystyle f(x)={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}}
The parameter


{\displaystyle \mu }
is the mean or expectation of the distribution (and also its median and mode), while the parameter


{\displaystyle \sigma }
is its standard deviation. The variance of the distribution is



{\displaystyle \sigma ^{2}}
. A random variable with a Gaussian distribution is said to be normally distributed, and is called a normal deviate.
Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known. Their importance is partly due to the central limit theorem. It states that, under some conditions, the average of many samples (observations) of a random variable with finite mean and variance is itself a random variable—whose distribution converges to a normal distribution as the number of samples increases. Therefore, physical quantities that are expected to be the sum of many independent processes, such as measurement errors, often have distributions that are nearly normal.Moreover, Gaussian distributions have some unique properties that are valuable in analytic studies. For instance, any linear combination of a fixed collection of normal deviates is a normal deviate. Many results and methods, such as propagation of uncertainty and least squares parameter fitting, can be derived analytically in explicit form when the relevant variables are normally distributed.
A normal distribution is sometimes informally called a bell curve. However, many other distributions are bell-shaped (such as the Cauchy, Student's t, and logistic distributions).

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  1. S

    Probability of Hypokalemia w/ 1 or Multiple Measurements

    TL;DR Summary: Finding the probability with one measurement and multiple measurements on separate days. Question: Hypokalemia is diagnosed when blood potassium levels are low, below 3.5 mmol/L. Let’s assume we know a patient whose measured potassium levels vary daily according to N(µ = 3.8...
  2. A

    A Discrete type normal distribution

    The following is given: $$\displaystyle P(K = k) = \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(k -\mu \right)^{2}}{\sigma ^{2}}}}{\sigma ~\sqrt{\pi }}$$ How can you prove that the following equalities are correct? $$\displaystyle \sum _{k=-\infty }^{\infty }1/2\,{\frac {...
  3. A

    A A different discrete normal distribution

    In the article A Discrete Normal Distribution of Dilip Roy in the journal COMMUNICATION IN STATISTICS Theory and methods Vol. 32, no. 10, pp. 1871-1883, 2003 one can read: A discrete normal (##dNormal##) variate, ##dX##, can be viewed as the discrete concentration of the normal variate ##X##...
  4. H

    Improper integral of a normal function

    I'm trying to solve an improper integral, but I'm not familiar with this kind of integral. ##\int_{-\infty}^{\infty} (xa^3 e^{-x^2} + ab e^{-x^2}) dx## a and b are both constants. From what I found ##\int_{-\infty}^{\infty} d e^{-u^2} dx = \sqrt{\pi}##, where d is a constant and...
  5. chwala

    Solve the problem that involves Normal distribution

    My interest is on part (c), My take, ##Z=\dfrac{160−200}{60}=−0.666666## ##Pr(−0.66666)=0.3546## ##⇒\dfrac{x_1-200}{60}=1.05## ##x_1=63+200=263## Yes, i am aware that they want the answer to ##5## significant figures...i just wanted to check the alternative method... Appreciate your insight...
  6. A

    A A discrete version of the normal distribution

    I have the following function for the normal distribution: $$\displaystyle f \left(x \right) = \frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -\mu \right)^{2}}{\sigma ^{2}}}}{\sigma ~\sqrt{\pi }}$$ How can the following integrals be equal to their sums? $$\displaystyle \int_{-\infty...
  7. A

    A The normal equivalent for a discrete random variable

    De normal distribution has the following form: $$\displaystyle f \left(x \right) \, = \,\frac{1}{2}~\frac{\sqrt{2}~e^{-\frac{1}{2}~\frac{\left(x -\nu \right)^{2}}{\tau ^{2}}}}{\tau ~\sqrt{\pi }}$$ and it's integral is equal to one: $$\displaystyle \int_{-\infty }^{\infty }\!1/2\,{\frac {...
  8. WMDhamnekar

    MHB How to prove the normal distribution tail inequality for large x ?

    What is the meaning of this proof? What is the meaning of last statement of this proof? How to prove lemma (7.1)? or How to answer problem 1 given below?
  9. karush

    MHB Normal distribution graph P(a<x<b)

    The probability that the lifespan of an insect of this species lies between 55 and 60 hours is represented by the shaded area in the following diagram.\\ Write down the values of a and b. $a=\dfrac{2}{4.4}= 0.455 b=\dfrac{3}{4.4}=0.682]$ ok this was a key to a test question from 2013 but mostly...
  10. Ackbach

    MHB Likelihood Ratio Test for Common Variance from Two Normal Distribution Samples

    $\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: Let $S_1^2$ and $S_2^2$ denote, respectively, the variances of independent random samples of sizes $n$ and $m$ selected from normal distributions with means $\mu_1$ and $\mu_2$ and common...
  11. S

    Probability related to Normal Distribution

    a) Let X = distance walked on Friday and Y = distance walked on Saturday X ~ N (12, 0.192) and Y ~ N (10, 0.52) Let A = Y - X → A ~ N (-2 , 0.2861) P(Y > X) = P(Y - X > 0) = P(A > 0) = 9.2 x 10-5 But the answer key is 0.026 Where is my mistake? Thanks
  12. S

    Why a normal distribution is not a good approximation for these exam scores?

    I am not really sure what the reason is but my argument would be if normal distribution is appropriate, then almost all the score will fall in the range of μ - 3σ to μ + 3σ For this case, the range of μ - 3σ to μ + 3σ is 26.6 to 118.4 and all the score is unlikely to be within the range. I...
  13. S

    B Question about Chi-Square Test Regarding Normal Distribution

    The first step is to group the data and make a table so I can get the observed frequency for each data interval. I did two different groupings (something like 150 - 160 , 160 - 170 , etc and the other is 150 - 170, 170 - 190, etc) and found out that the conclusion of the hypothesis is different...
  14. D

    Three independent random variables having Normal distribution

    Let ##X_1 X_2 X_3 ## be three independent random variables having Normal(Gaussian ) distribution, all with mean ##\mu##=20 and variance ##\sigma^2##=9. Also let ##S=X_1+ X_2 +X_3## and let ##N## be the number of the ##X_i## assuming values greater than 25. ##E\left[N\right]##=? I did not...
  15. N

    I Stern-Gerlach w. normal distribution if magnets were more separated?

    Just wondered if the power of mags. is decreased, or they are more separated, don't you get a normal distribution ? (I'm in biology) - would you also not have predicted that w. reasonably strong magnets, they will either end one one side or the other ? Thx a lot!
  16. J

    MHB Weak Convergence to Normal Distribution

    Problem: Let $X_n$ be independent random variables such that $X_1 = 1$, and for $n \geq 2$, $P(X_n=n)=n^{-2}$ and $P(X_n=1)=P(X_n=0)=\frac{1}{2}(1-n^{-2})$. Show $(1/\sqrt{n})(\sum_{m=1}^{n}X_n-n/2)$ converges weakly to a normal distribution as $n \rightarrow \infty$.Thoughts: My professor...
  17. PainterGuy

    B Questions about a normal distribution (discrete to continuous)

    Hi, I was watching this Youtube video (please remove the parentheses) : https://youtu.(be/mtH1fmUVkfE?t=215) While watching it, a question came to my mind. In the picture, you can easily calculate the total number of customers. It's 1000. For my question, I'm going to use the same picture...
  18. N

    I Variable transformation for a multivariate normal distribution

    Hello. I would like to draw (sample) several random vectors x from a n-dimensional multivariate normal distribution. For this purpose I want to use C++ and the GNU Scientific Library function gsl_ran_multivariate_gaussian ...
  19. fisher garry

    I Bivariate normal distribution from normal linear combination

    I can't prove this proposition. I have however managed to prove that the linear combinations of the independent normal rv's are also normal by looking at it's mgf $$E(e^{X_1+X_2+...+X_n})=E(e^{X_1})E(e^{X_2})...E(e^{X_n})$$ The mgf of a normal distribution is $$e^{\mu t}e^{\frac{t^2...
  20. Hiero

    Integral of a normal distribution

    Homework Statement (Scroll to bottom for the true question) Suppose we are to find the integral from -∞ to +∞ of (let’s just say) e-x2dx Homework Equations ∫∫f(x)g(y)dxdy = (∫f(x)dx)(∫g(y)dy) The Attempt at a Solution We can square the integral we want to solve for then use my relevant...
  21. D

    MHB Expression for normal distribution

    Write an expression for normal distribution for the data: Measured values are: 4,393; 4,372; 4,381; 4,373 and 4,401 thanks for help
  22. H

    I Why does normal distribution turn into t distribution when variance is unknown?

    Suppose ##X## ~ N(##\mu##,##\sigma^2##). Then ##\bar{X}## ~ N(##\mu##,##\frac{\sigma^2}{n}##), where ##\bar{X}## is the random variable for sample mean for samples of size ##n##. But when the population variance ##\sigma^2## is unknown and the sample size ##n## is small, ##\bar{X}## no longer...
  23. D

    A Estimation error from estimation quantile of normal distribution

    Hi guys, For my (master) project I am trying to find the probability that a random variable, which is normally distributed, exceeds a quantile that is estimated by a limited number of observations. See attached for my attempt. - Is it correct? - How to incorporate the fact that the mean and...
  24. Danny Boy

    A Defining Krauss operators with normal distribution

    I am interested in defining Krauss operators which allow you to define quantum measurements peaked at some basis state. To this end I am considering the Normal Distribution. Consider a finite set of basis states ##\{ |x \rangle\}_x## and a set of quantum measurement operators of the form $$A_C =...
  25. T

    I Proving a multivariate normal distribution by the moment generating function

    I have proved (8.1). However I am trying to prove that ##\bar{X},X_i-\bar{X},i=1,...,n## has a joint distribution that is multivariate normal. I am trying to prove it by looking at the moment generating function: ##E(e^{t(X_i-\bar{X})}=E(e^{tX_i})E(e^{-\frac{t}{n}\sum_{i=1}^n X_i})## I am...
  26. P

    MHB James' question about Normal Distribution

    (a) We are told $\displaystyle \begin{align*} \textrm{Pr}\,\left( X < 3 \right) = \textrm{Pr}\,\left( Z < a \right) \end{align*}$, so if $\displaystyle \begin{align*} x = 3 \end{align*}$ and $\displaystyle \begin{align*} z = a \end{align*}$ then we have $\displaystyle \begin{align*} z &=...
  27. O

    Normal Distribution Question

    Homework Statement Suppose that X ~ N(μ,σ). Find a in terms of μ and σ if P(X>a) = 1/3 * P(X ≤a) Homework EquationsThe Attempt at a Solution 1 - P(X ≤a) = 1/3 * P(X ≤a) 3 = 4P(X ≤a) P(X ≤a) = 3/4 Since x0 = μ + σz0 where x0 and z0 are the same percentile for N(μ,σ) and N(0,1)...
  28. jdawg

    Statistics: Standard Deviation for a Normal Distribution

    Homework Statement A company allows a maximum failure rate of 1 out of 250,000 parts. To insure this quality goal, failed parts must be how many standard deviations from the mean? Use Excel to solve. Homework Equations z= (X-μ)/σ The Attempt at a Solution Hi! So I'm assuming that this is a...
  29. Z

    75th percentile of the normal distribution

    Homework Statement The random variable X is normally distributed. The values 650 and 850 are at the 60th and 90th percentile of the distribution of X respectively. Find the value of the 75th percentile of the distribution of X. Homework Equations I found the above formula from internet. where...
  30. B

    MHB Normal Distribution: Percentiles

    The number of chocolate chips in a bag of chocolate chip cookies is approximately normally distributed with a mean of 1263 chips and a standard deviation of 117chips. ​(a) Determine the 26th percentile for the number of chocolate chips in a bag. ​(b) Determine the number of chocolate chips in...
  31. R

    I The Normal Distribution - Random Errors

    So let's say I do some measurements and obtain a set of measured values. The measurement is characterized by random errors so by making enough measurements, they approach a normal distribution. In other words, my set of measured values can be approximated by a normal distribution characterized...
  32. M

    MHB Height : Normal distribution

    Hey! :o I am looking at the following: The average tallest men live in Netherlands and Montenegro mit $1.83$m=$183$cm. The average shortest men live in Indonesia mit $1.58$m=$158$cm. The standard deviation of the height in Netherlands/Montenegro is $9.7$cm and in Indonesia it is $7.8$cm...
  33. T

    B Probability density of a normal distribution

    If the normalized probability density of the normal distribution is ## p(x) = \frac {1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}} ##, then if ##\sigma = 0.0001## and in the special case ## x = \mu##, wouldn't the probability density at this point, ##p(\mu)##, exceed 1 since it is equal...
  34. senobim

    A Moments of normal distribution

    I have calculated characteristic function of normal distribution f_{X}(k)=e^{(ika-\frac{\sigma ^{2}k^{2}}{2})} and now I would like to find the moments, so I know that you could expand characteristic function by Taylor series f_{X}(k)=exp(1+\frac{1}{1!}(ika -...
  35. W

    MHB Median, mode, normal distribution

    In a digital communication channel, assume that the number of bits received in error can be modeled by a binomial random variable, and assumed that a bit is received in error is 1×〖10〗^(−5) . if 16 million bits are transmitted, What is the probability that more than 150 errors occur? Find the...
  36. W

    MHB Mean & Std Dev for Norm Dist. Exam Marks - 450 Stud.

    Assuming that the number of marks scored by a candidate is normally distributed, find the mean and the standard deviation, if the number of first class students(60% or more marks) is 25, the number of failed students(less than 30%marks) is 90 and the total number of candidates appearing for the...
  37. Avatrin

    I Understanding Mahalanobis distance

    I am currently taking a course in pattern recognition, and several times I have encountered the multivariable normal distribution and thus, Mahalanobis distance. I want to understand Mahalanobis distance; Primarily for understanding the normal distribution, but also to understand the measure...
  38. J

    MHB Statistics Normal Distribution

    Eleanor scores 680 on the mathematics part of the SAT. The distribution of SAT math scores in recent years has been Normal with mean 547 and standard deviation 85. Gerald takes the ACT Assessment mathematics test and scores 27. ACT math scores are Normally distributed with mean 21.3 and...
  39. perplexabot

    A Derivative of log of normal distribution

    Hey all, I've had this point of confusion for a bit and I have thought that with time I may be able to clear it out myself. Nope, hasn't happened. I think I need help. Let us say we have the following \phi_{k+1}=\phi_{k}+v_k where, v_k\overset{iid}{\sim}\mathcal{N}(0,\sigma^2) and...
  40. H

    Normal Distribution Question. Need help

    [MENTOR note] Post moved from General Math forum hence no template. Assume that a random variable follows a normal distribution with a mean of 80 and a standard deviation of 24. What percentage of this distribution is not between 32 and 116? My approach is to calculate the Probability for (mean...
  41. chwala

    Problem on normal distribution

    Homework Statement The time Rafa spends on his homework each day is normally distributed with mean 1.9hrs and standard deviation σ. On 80% of these days he spends more than 1.35 hours on his homework. i. find the value of σ ii. find the probability that Rafa spends less than 2 hours on his...
  42. Erland

    I Why is so much well described by the normal distribution?

    Why are so many phenomena well described by the normal distribution? For example: the height of 18 year old males in Sweden, the weight of apples on a particular tree, the volume of coke cans (supposed to be 33 cl), etc. etc. are all well described by the normal distribution. How come? A...
  43. R

    A Deriving the standard normal distribution

    I've calculated the joint distribution, XY_PDF(x,y) of random variables X and Y (both coming from a distribution N(n) = C*e^(-K*n^2)). I use XY_PDF(x,y) to calculate the joint distribution AR_PDF(a,r) of the random variables A (angle) and R (radius), with the PDF method and the Jacobian. Since...
  44. NihalRi

    Calculating Variance in Normal Distribution with Given Percentile

    Homework Statement A student sits a test and is told that the marks follow a normal distribution with mean 100. The student receives a mark of 124 and is told that he is at the 68th percentile. Calculate the variance of the distribution. Homework Equations...
  45. Amcote

    Stats: Approximating a binomial with a normal distribution

    Homework Statement A multiple choice test consists of a series of questions, each with four possible answers. How many questions are needed in order to be 99% confident that a student who guesses blindly at each question scores no more than 35% on the test? Homework Equations So I know that...
  46. J

    I Normal distribution and constant variance

    Why do people say that RVs that have the normal distribution has a constant variance. What does that mean constant variance.
  47. Q

    Continous limit of a multivariate normal distribution

    Hello everyone, I am currently considering a set of random variables, \vec{x} = [x_1,x_2,...x_N] which are know to follow a multivariate normal distribution, P(\vec{x}) \propto \mathrm{exp}(-\frac{1}{2}(\vec{x}-\vec{\mu})^\mathrm{T}\Sigma^{-1}(\vec{x}-\vec{\mu})) The covariance matrix Σ and...
  48. S

    Normal distribution curve area?

    Is there a relatively simple algorithm to compute the area in percentage under the curve as represented by a sigma value? For example; 3 sigma = 99.7 2 sigma = 95 1 sigma = 68.3 Now suppose I wanted to know 2.5 sigma without a table.
  49. T

    A normal distribution of IQ scores

    Homework Statement It is known that the IQ score of ten-year-old children in a particular population has a normal distribution with mean 100 and standard deviation 15. (a) What proportion of this population have an IQ score above 115? (b) Mary’s IQ is equal to the 80th percentile of this...
  50. T

    MHB Standard normal distribution probability

    "A study of long distance phone calls made from the corporate offices of the Pepsi Bottling Group Inc. showed the calls follow the normal distribution. The mean length of time per call was 4.2 minutes and the standard deviation was 0.60 minutes. a.What is the probability the calls lasted...