10: The Normal (Gaussian) Distribution
Lisa Yan, CS109, 2020 Carl Friedrich Gauss Carl Friedrich Gauss (1777-1855) was a remarkably influential German mathematician Did not invent Normal distribution but rather popularized it
The Normal Distribution - Stanford University
Normal random variable An normal (= Gaussian) random variable is a good approximation to many other distributions It often results from sums or averages of independent random variables X∼N(μ,σ2) fX(x)= 1 σ√2π e − 1 2(x−μ σ) 2
Normal distribution - University of Notre Dame
Normal distribution The normal distribution is the most widely known and used of all distributions Because the normal distribution approximates many natural phenomena so well, it has developed into a standard of reference for many probability problems I Characteristics of the Normal distribution • Symmetric, bell shaped
Tables of Normal Values (As of January 2013)
Tables of Normal Values (As of January 2013) Note: Values and units of measurement listed in these tables are derived from several resources Substantial variation exists in the ranges quoted as “normal” and may vary depending on the assay used by different laboratories Therefore, these tables should be considered as directional only
Standard Normal Distribution Table - SOA
STANDARD NORMAL DISTRIBUTION TABLE Entries represent Pr(Z ≤ z) The value of z to the first decimal is given in the left column The second decimal is given in the top row
Normal Lab Values Chart - IM 2015
Normal Male — 130 mL/min/1 73 m2 Female — 120 mL/min/1 73 m2 Stages of Chronic Kidney Disease Stage 1 — greater than or equal to 90 mL/min/1 73 m2
The Conjugate Prior for the Normal Distribution
2 The Conjugate Prior for the Normal Distribution Remark 3 These formulas are extremely useful so you should memorize them They are easily derived based on the notion of a Schur complement of a matrix We apply this lemma with the correspondence: xz 2, z 1 x= + ˙" "˘N(0;1) = 0 + ˙ 0 ˘N(0;1) E(x) = 0 (5)
Normal Gait - MU School of Medicine
Normal Gait Heikki Uustal, MD Medical Director, Prosthetic/Orthotic Team JFK-Johnson Rehab Institute Edison, NJ 1
[PDF] indice de masse corporelle
[PDF] calculer son imc
[PDF] cdt alcool taux normal
[PDF] taux cdt
[PDF] calcul taux alcoolémie formule
[PDF] taux d'alcoolémie mortel
[PDF] taux d'alcool permis quebec
[PDF] 0.08 alcool age
[PDF] multiplication et division de fraction
[PDF] évaluation 5ème maths
[PDF] mathématiques 5ème
[PDF] cours maths 5ème
[PDF] la sphère terrestre correction
[PDF] la sphere terrestre dm
Normal distribution
The normal distribution is the most widely known and used of all distributions. Because the normal distribution approximates many natural phenomena so well, it has developed into a standard of reference for ma ny probability problems.I. Characteristics of the Normal distribution
Symmetric, bell shaped
Continuous f
or all values of X between - and so that each conceivable interval of real numbers has a probability other than zero. - X Two parameters, µ and ı. Note that the normal distribution is actually a family of distributions, since µ and ı determine the shape of the distribution.The rule for a normal density function is e
2 1 = ) , f(x; 22/2)--(x 2 2
The notation N(µ, ı
2 ) means normally distributed with mean µ and variance ı 2 . If we sayX N(µ, ı
2 ) we mean that X is distributed N(µ, ı 2 About 2/3 of all cases fall within one standard deviation of the mean, that isP(µ - ı X µ + ı) = .6826.
About 95% of cases lie within 2 standard deviations of the mean, that isP(µ - 2ı X µ + 2ı) = .9544
Normal distribution - Page 1
II. Why is the normal distribution useful?
Many things actually are normally distributed, or very close to it. For example, height and intelligence are approximately normally distributed; measurement errors also often have a normal distribution The normal distribution is easy to work with mathematically. In many practical cases, the methods developed using normal theory work quite well even when the distribution is not normal. There is a very strong connection between the size of a sample N and the extent to which a sampling distribution approaches the normal form. Many sampling distributions based on large N can be approximated by the normal distribution even though the population distribution itself is definitely not normal.III. The standardized normal distribution.
a. General Procedure. As you might suspect from the formula for the normal density function, it would be difficult and tedious to do the calculus e very time we had a new set of parameters for and . So instead, we usually work with the standardized normaldistribution, where µ = 0 and ı = 1, i.e. N(0,1). That is, rather than directly solve a problem
involving a normally distributed variable X with mean µ and standard deviation ı, an indirect approach is used.1. We first convert the problem into an equivalent one dealing with a normal
variable measured in standardized deviation units, called a standardized normal variable. To do this, if X N(µ, ı5), then1) N(0,
- X = Z~2. A table of standardized normal values (Appendix E, Table I) can then be
used to obtain an answer in terms of the converted problem.3. If necessary, we can then convert back to the original units of
measurement. To do this, simply note that, if we take the formula for Z, multiply both sides byı, and then add µ to both sides, we get
+ Z= X4. The interpetation of Z values is straightforward. Since = 1, if Z = 2, the
corresponding X value is exactly 2 standard deviations above the mean. If Z = -1, the corresponding X value is one standard deviation below the mean. If Z = 0, X = the mean, i.e. . b. Rules for using the standardized normal distribution. It is very important to understand how the standardized normal distribution works, so we will spend some time here going over it. Recall that, for a random variable X,F(x) = P(X x)
Normal distribution - Page 2
Appendix E, Table I (Or see Hays, p. 924) reports the cumulative normal probabilities fornormally distributed variables in standardized form (i.e. Z-scores). That is, this table reports P(Z
z) = F(z). For a given value of Z, the table reports what proportio n of the distribution lies below that value. For example, F(0) = .5; half the area of the standardized normal curve lies to the left of Z = 0. Note that only positive values of Z are reported; as we will see, this is not a problem, since the normal distribution is symmetric. We will now show how to work with this table. NOTE: While memorization may be useful, you will be much better off if you gain an intuitive understanding as to why the rules that follow are correct. Tr y drawing pictures of the normal distribution to convince yourself that each rule is valid.