The Mean, Variance and Covariance
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Lecture 22: Bivariate Normal Distribution Distribution
6.5 Conditional Distributions General Bivariate Normal Let Z1; Z2 ∼ N (0; 1), which we will use to build a general bivariate normal Lecture 22: Bivariate Normal Distribution distribution. 1 1 2 2 f (z1; z2) = exp − (z1 + z2 ) Statistics 104 2π 2 We want to transform these unit normal distributions to have the follow Colin Rundel arbitrary parameters: µX ; µY ; σX ; σY ; ρ April 11, 2012 X = σX Z1 + µX p 2 Y = σY [ρZ1 + 1 − ρ Z2] + µY Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 1 / 22 6.5 Conditional Distributions 6.5 Conditional Distributions General Bivariate Normal - Marginals General Bivariate Normal - Cov/Corr First, lets examine the marginal distributions of X and Y , Second, we can find Cov(X ; Y ) and ρ(X ; Y ) Cov(X ; Y ) = E [(X − E(X ))(Y − E(Y ))] X = σX Z1 + µX h p i = E (σ Z + µ − µ )(σ [ρZ + 1 − ρ2Z ] + µ − µ ) = σX N (0; 1) + µX X 1 X X Y 1 2 Y Y 2 h p 2 i = N (µX ; σX ) = E (σX Z1)(σY [ρZ1 + 1 − ρ Z2]) h 2 p 2 i = σX σY E ρZ1 + 1 − ρ Z1Z2 p 2 2 Y = σY [ρZ1 + 1 − ρ Z2] + µY = σX σY ρE[Z1 ] p 2 = σX σY ρ = σY [ρN (0; 1) + 1 − ρ N (0; 1)] + µY = σ [N (0; ρ2) + N (0; 1 − ρ2)] + µ Y Y Cov(X ; Y ) ρ(X ; Y ) = = ρ = σY N (0; 1) + µY σX σY 2 = N (µY ; σY ) Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 2 / 22 Statistics 104 (Colin Rundel) Lecture 22 April 11, 2012 3 / 22 6.5 Conditional Distributions 6.5 Conditional Distributions General Bivariate Normal - RNG Multivariate Change of Variables Consequently, if we want to generate a Bivariate Normal random variable Let X1;:::; Xn have a continuous joint distribution with pdf f defined of S. -
Applied Biostatistics Mean and Standard Deviation the Mean the Median Is Not the Only Measure of Central Value for a Distribution
Health Sciences M.Sc. Programme Applied Biostatistics Mean and Standard Deviation The mean The median is not the only measure of central value for a distribution. Another is the arithmetic mean or average, usually referred to simply as the mean. This is found by taking the sum of the observations and dividing by their number. The mean is often denoted by a little bar over the symbol for the variable, e.g. x . The sample mean has much nicer mathematical properties than the median and is thus more useful for the comparison methods described later. The median is a very useful descriptive statistic, but not much used for other purposes. Median, mean and skewness The sum of the 57 FEV1s is 231.51 and hence the mean is 231.51/57 = 4.06. This is very close to the median, 4.1, so the median is within 1% of the mean. This is not so for the triglyceride data. The median triglyceride is 0.46 but the mean is 0.51, which is higher. The median is 10% away from the mean. If the distribution is symmetrical the sample mean and median will be about the same, but in a skew distribution they will not. If the distribution is skew to the right, as for serum triglyceride, the mean will be greater, if it is skew to the left the median will be greater. This is because the values in the tails affect the mean but not the median. Figure 1 shows the positions of the mean and median on the histogram of triglyceride. -
1. How Different Is the T Distribution from the Normal?
Statistics 101–106 Lecture 7 (20 October 98) c David Pollard Page 1 Read M&M §7.1 and §7.2, ignoring starred parts. Reread M&M §3.2. The eects of estimated variances on normal approximations. t-distributions. Comparison of two means: pooling of estimates of variances, or paired observations. In Lecture 6, when discussing comparison of two Binomial proportions, I was content to estimate unknown variances when calculating statistics that were to be treated as approximately normally distributed. You might have worried about the effect of variability of the estimate. W. S. Gosset (“Student”) considered a similar problem in a very famous 1908 paper, where the role of Student’s t-distribution was first recognized. Gosset discovered that the effect of estimated variances could be described exactly in a simplified problem where n independent observations X1,...,Xn are taken from (, ) = ( + ...+ )/ a normal√ distribution, N . The sample mean, X X1 Xn n has a N(, / n) distribution. The random variable X Z = √ / n 2 2 Phas a standard normal distribution. If we estimate by the sample variance, s = ( )2/( ) i Xi X n 1 , then the resulting statistic, X T = √ s/ n no longer has a normal distribution. It has a t-distribution on n 1 degrees of freedom. Remark. I have written T , instead of the t used by M&M page 505. I find it causes confusion that t refers to both the name of the statistic and the name of its distribution. As you will soon see, the estimation of the variance has the effect of spreading out the distribution a little beyond what it would be if were used. -
04 – Everything You Want to Know About Correlation but Were
EVERYTHING YOU WANT TO KNOW ABOUT CORRELATION BUT WERE AFRAID TO ASK F R E D K U O 1 MOTIVATION • Correlation as a source of confusion • Some of the confusion may arise from the literary use of the word to convey dependence as most people use “correlation” and “dependence” interchangeably • The word “correlation” is ubiquitous in cost/schedule risk analysis and yet there are a lot of misconception about it. • A better understanding of the meaning and derivation of correlation coefficient, and what it truly measures is beneficial for cost/schedule analysts. • Many times “true” correlation is not obtainable, as will be demonstrated in this presentation, what should the risk analyst do? • Is there any other measures of dependence other than correlation? • Concordance and Discordance • Co-monotonicity and Counter-monotonicity • Conditional Correlation etc. 2 CONTENTS • What is Correlation? • Correlation and dependence • Some examples • Defining and Estimating Correlation • How many data points for an accurate calculation? • The use and misuse of correlation • Some example • Correlation and Cost Estimate • How does correlation affect cost estimates? • Portfolio effect? • Correlation and Schedule Risk • How correlation affect schedule risks? • How Shall We Go From Here? • Some ideas for risk analysis 3 POPULARITY AND SHORTCOMINGS OF CORRELATION • Why Correlation Is Popular? • Correlation is a natural measure of dependence for a Multivariate Normal Distribution (MVN) and the so-called elliptical family of distributions • It is easy to calculate analytically; we only need to calculate covariance and variance to get correlation • Correlation and covariance are easy to manipulate under linear operations • Correlation Shortcomings • Variances of R.V. -
18.440: Lecture 9 Expectations of Discrete Random Variables
18.440: Lecture 9 Expectations of discrete random variables Scott Sheffield MIT 18.440 Lecture 9 1 Outline Defining expectation Functions of random variables Motivation 18.440 Lecture 9 2 Outline Defining expectation Functions of random variables Motivation 18.440 Lecture 9 3 Expectation of a discrete random variable I Recall: a random variable X is a function from the state space to the real numbers. I Can interpret X as a quantity whose value depends on the outcome of an experiment. I Say X is a discrete random variable if (with probability one) it takes one of a countable set of values. I For each a in this countable set, write p(a) := PfX = ag. Call p the probability mass function. I The expectation of X , written E [X ], is defined by X E [X ] = xp(x): x:p(x)>0 I Represents weighted average of possible values X can take, each value being weighted by its probability. 18.440 Lecture 9 4 Simple examples I Suppose that a random variable X satisfies PfX = 1g = :5, PfX = 2g = :25 and PfX = 3g = :25. I What is E [X ]? I Answer: :5 × 1 + :25 × 2 + :25 × 3 = 1:75. I Suppose PfX = 1g = p and PfX = 0g = 1 − p. Then what is E [X ]? I Answer: p. I Roll a standard six-sided die. What is the expectation of number that comes up? 1 1 1 1 1 1 21 I Answer: 6 1 + 6 2 + 6 3 + 6 4 + 6 5 + 6 6 = 6 = 3:5. 18.440 Lecture 9 5 Expectation when state space is countable II If the state space S is countable, we can give SUM OVER STATE SPACE definition of expectation: X E [X ] = PfsgX (s): s2S II Compare this to the SUM OVER POSSIBLE X VALUES definition we gave earlier: X E [X ] = xp(x): x:p(x)>0 II Example: toss two coins. -
11. Correlation and Linear Regression
11. Correlation and linear regression The goal in this chapter is to introduce correlation and linear regression. These are the standard tools that statisticians rely on when analysing the relationship between continuous predictors and continuous outcomes. 11.1 Correlations In this section we’ll talk about how to describe the relationships between variables in the data. To do that, we want to talk mostly about the correlation between variables. But first, we need some data. 11.1.1 The data Table 11.1: Descriptive statistics for the parenthood data. variable min max mean median std. dev IQR Dan’s grumpiness 41 91 63.71 62 10.05 14 Dan’s hours slept 4.84 9.00 6.97 7.03 1.02 1.45 Dan’s son’s hours slept 3.25 12.07 8.05 7.95 2.07 3.21 ............................................................................................ Let’s turn to a topic close to every parent’s heart: sleep. The data set we’ll use is fictitious, but based on real events. Suppose I’m curious to find out how much my infant son’s sleeping habits affect my mood. Let’s say that I can rate my grumpiness very precisely, on a scale from 0 (not at all grumpy) to 100 (grumpy as a very, very grumpy old man or woman). And lets also assume that I’ve been measuring my grumpiness, my sleeping patterns and my son’s sleeping patterns for - 251 - quite some time now. Let’s say, for 100 days. And, being a nerd, I’ve saved the data as a file called parenthood.csv. -
LECTURES 2 - 3 : Stochastic Processes, Autocorrelation Function
LECTURES 2 - 3 : Stochastic Processes, Autocorrelation function. Stationarity. Important points of Lecture 1: A time series fXtg is a series of observations taken sequentially over time: xt is an observation recorded at a specific time t. Characteristics of times series data: observations are dependent, become available at equally spaced time points and are time-ordered. This is a discrete time series. The purposes of time series analysis are to model and to predict or forecast future values of a series based on the history of that series. 2.2 Some descriptive techniques. (Based on [BD] x1.3 and x1.4) ......................................................................................... Take a step backwards: how do we describe a r.v. or a random vector? ² for a r.v. X: 2 d.f. FX (x) := P (X · x), mean ¹ = EX and variance σ = V ar(X). ² for a r.vector (X1;X2): joint d.f. FX1;X2 (x1; x2) := P (X1 · x1;X2 · x2), marginal d.f.FX1 (x1) := P (X1 · x1) ´ FX1;X2 (x1; 1) 2 2 mean vector (¹1; ¹2) = (EX1; EX2), variances σ1 = V ar(X1); σ2 = V ar(X2), and covariance Cov(X1;X2) = E(X1 ¡ ¹1)(X2 ¡ ¹2) ´ E(X1X2) ¡ ¹1¹2. Often we use correlation = normalized covariance: Cor(X1;X2) = Cov(X1;X2)=fσ1σ2g ......................................................................................... To describe a process X1;X2;::: we define (i) Def. Distribution function: (fi-di) d.f. Ft1:::tn (x1; : : : ; xn) = P (Xt1 · x1;:::;Xtn · xn); i.e. this is the joint d.f. for the vector (Xt1 ;:::;Xtn ). (ii) First- and Second-order moments. ² Mean: ¹X (t) = EXt 2 2 2 2 ² Variance: σX (t) = E(Xt ¡ ¹X (t)) ´ EXt ¡ ¹X (t) 1 ² Autocovariance function: γX (t; s) = Cov(Xt;Xs) = E[(Xt ¡ ¹X (t))(Xs ¡ ¹X (s))] ´ E(XtXs) ¡ ¹X (t)¹X (s) (Note: this is an infinite matrix). -
5. the Student T Distribution
Virtual Laboratories > 4. Special Distributions > 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 5. The Student t Distribution In this section we will study a distribution that has special importance in statistics. In particular, this distribution will arise in the study of a standardized version of the sample mean when the underlying distribution is normal. The Probability Density Function Suppose that Z has the standard normal distribution, V has the chi-squared distribution with n degrees of freedom, and that Z and V are independent. Let Z T= √V/n In the following exercise, you will show that T has probability density function given by −(n +1) /2 Γ((n + 1) / 2) t2 f(t)= 1 + , t∈ℝ ( n ) √n π Γ(n / 2) 1. Show that T has the given probability density function by using the following steps. n a. Show first that the conditional distribution of T given V=v is normal with mean 0 a nd variance v . b. Use (a) to find the joint probability density function of (T,V). c. Integrate the joint probability density function in (b) with respect to v to find the probability density function of T. The distribution of T is known as the Student t distribution with n degree of freedom. The distribution is well defined for any n > 0, but in practice, only positive integer values of n are of interest. This distribution was first studied by William Gosset, who published under the pseudonym Student. In addition to supplying the proof, Exercise 1 provides a good way of thinking of the t distribution: the t distribution arises when the variance of a mean 0 normal distribution is randomized in a certain way. -
Randomized Algorithms
Chapter 9 Randomized Algorithms randomized The theme of this chapter is madendrizo algorithms. These are algorithms that make use of randomness in their computation. You might know of quicksort, which is efficient on average when it uses a random pivot, but can be bad for any pivot that is selected without randomness. Analyzing randomized algorithms can be difficult, so you might wonder why randomiza- tion in algorithms is so important and worth the extra effort. Well it turns out that for certain problems randomized algorithms are simpler or faster than algorithms that do not use random- ness. The problem of primality testing (PT), which is to determine if an integer is prime, is a good example. In the late 70s Miller and Rabin developed a famous and simple random- ized algorithm for the problem that only requires polynomial work. For over 20 years it was not known whether the problem could be solved in polynomial work without randomization. Eventually a polynomial time algorithm was developed, but it is much more complicated and computationally more costly than the randomized version. Hence in practice everyone still uses the randomized version. There are many other problems in which a randomized solution is simpler or cheaper than the best non-randomized solution. In this chapter, after covering the prerequisite background, we will consider some such problems. The first we will consider is the following simple prob- lem: Question: How many comparisons do we need to find the top two largest numbers in a sequence of n distinct numbers? Without the help of randomization, there is a trivial algorithm for finding the top two largest numbers in a sequence that requires about 2n − 3 comparisons. -
Construct Validity and Reliability of the Work Environment Assessment Instrument WE-10
International Journal of Environmental Research and Public Health Article Construct Validity and Reliability of the Work Environment Assessment Instrument WE-10 Rudy de Barros Ahrens 1,*, Luciana da Silva Lirani 2 and Antonio Carlos de Francisco 3 1 Department of Business, Faculty Sagrada Família (FASF), Ponta Grossa, PR 84010-760, Brazil 2 Department of Health Sciences Center, State University Northern of Paraná (UENP), Jacarezinho, PR 86400-000, Brazil; [email protected] 3 Department of Industrial Engineering and Post-Graduation in Production Engineering, Federal University of Technology—Paraná (UTFPR), Ponta Grossa, PR 84017-220, Brazil; [email protected] * Correspondence: [email protected] Received: 1 September 2020; Accepted: 29 September 2020; Published: 9 October 2020 Abstract: The purpose of this study was to validate the construct and reliability of an instrument to assess the work environment as a single tool based on quality of life (QL), quality of work life (QWL), and organizational climate (OC). The methodology tested the construct validity through Exploratory Factor Analysis (EFA) and reliability through Cronbach’s alpha. The EFA returned a Kaiser–Meyer–Olkin (KMO) value of 0.917; which demonstrated that the data were adequate for the factor analysis; and a significant Bartlett’s test of sphericity (χ2 = 7465.349; Df = 1225; p 0.000). ≤ After the EFA; the varimax rotation method was employed for a factor through commonality analysis; reducing the 14 initial factors to 10. Only question 30 presented commonality lower than 0.5; and the other questions returned values higher than 0.5 in the commonality analysis. Regarding the reliability of the instrument; all of the questions presented reliability as the values varied between 0.953 and 0.956. -
1 One Parameter Exponential Families
1 One parameter exponential families The world of exponential families bridges the gap between the Gaussian family and general dis- tributions. Many properties of Gaussians carry through to exponential families in a fairly precise sense. • In the Gaussian world, there exact small sample distributional results (i.e. t, F , χ2). • In the exponential family world, there are approximate distributional results (i.e. deviance tests). • In the general setting, we can only appeal to asymptotics. A one-parameter exponential family, F is a one-parameter family of distributions of the form Pη(dx) = exp (η · t(x) − Λ(η)) P0(dx) for some probability measure P0. The parameter η is called the natural or canonical parameter and the function Λ is called the cumulant generating function, and is simply the normalization needed to make dPη fη(x) = (x) = exp (η · t(x) − Λ(η)) dP0 a proper probability density. The random variable t(X) is the sufficient statistic of the exponential family. Note that P0 does not have to be a distribution on R, but these are of course the simplest examples. 1.0.1 A first example: Gaussian with linear sufficient statistic Consider the standard normal distribution Z e−z2=2 P0(A) = p dz A 2π and let t(x) = x. Then, the exponential family is eη·x−x2=2 Pη(dx) / p 2π and we see that Λ(η) = η2=2: eta= np.linspace(-2,2,101) CGF= eta**2/2. plt.plot(eta, CGF) A= plt.gca() A.set_xlabel(r'$\eta$', size=20) A.set_ylabel(r'$\Lambda(\eta)$', size=20) f= plt.gcf() 1 Thus, the exponential family in this setting is the collection F = fN(η; 1) : η 2 Rg : d 1.0.2 Normal with quadratic sufficient statistic on R d As a second example, take P0 = N(0;Id×d), i.e. -
Random Variables and Probability Distributions 1.1
RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS 1. DISCRETE RANDOM VARIABLES 1.1. Definition of a Discrete Random Variable. A random variable X is said to be discrete if it can assume only a finite or countable infinite number of distinct values. A discrete random variable can be defined on both a countable or uncountable sample space. 1.2. Probability for a discrete random variable. The probability that X takes on the value x, P(X=x), is defined as the sum of the probabilities of all sample points in Ω that are assigned the value x. We may denote P(X=x) by p(x) or pX (x). The expression pX (x) is a function that assigns probabilities to each possible value x; thus it is often called the probability function for the random variable X. 1.3. Probability distribution for a discrete random variable. The probability distribution for a discrete random variable X can be represented by a formula, a table, or a graph, which provides pX (x) = P(X=x) for all x. The probability distribution for a discrete random variable assigns nonzero probabilities to only a countable number of distinct x values. Any value x not explicitly assigned a positive probability is understood to be such that P(X=x) = 0. The function pX (x)= P(X=x) for each x within the range of X is called the probability distribution of X. It is often called the probability mass function for the discrete random variable X. 1.4. Properties of the probability distribution for a discrete random variable.