DOCSLIB.ORG

Continuous Random Variables

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Continuous Random Variables ST 380 Probability and Statistics for the Physical Sciences Continuous Random Variables Recall: A continuous random variable X satisfies: 1 its range is the union of one or more real number intervals; 2 P(X = c) = 0 for every c in the range of X . Examples: The depth of a lake at a randomly chosen location. The pH of a random sample of effluent. The precipitation on a randomly chosen day is not a continuous random variable: its range is [0; 1), and P(X = c) = 0 for any c > 0, but P(X = 0) > 0. 1 / 12 Continuous Random Variables Probability Density Function ST 380 Probability and Statistics for the Physical Sciences Discretized Data Suppose that we measure the depth of the lake, but round the depth off to some unit. The rounded value Y is a discrete random variable; we can display its probability mass function as a bar graph, because each mass actually represents an interval of values of X . In R source("discretize.R") discretize(0.5) 2 / 12 Continuous Random Variables Probability Density Function ST 380 Probability and Statistics for the Physical Sciences As the rounding unit becomes smaller, the bar graph more accurately represents the continuous distribution: discretize(0.25) discretize(0.1) When the rounding unit is very small, the bar graph approximates a smooth function: discretize(0.01) plot(f, from = 1, to = 5) 3 / 12 Continuous Random Variables Probability Density Function ST 380 Probability and Statistics for the Physical Sciences The probability that X is between two values a and b, P(a ≤ X ≤ b), can be approximated by P(a ≤ Y ≤ b). Because Y is discrete, P(a ≤ Y ≤ b) is the sum of the areas of the corresponding bars in the graph. As the rounding unit becomes smaller, the sum of the areas of the bars approaches the integral of the smooth function. In the limit, Z b P(a ≤ X ≤ b) = f (x) dx: a 4 / 12 Continuous Random Variables Probability Density Function ST 380 Probability and Statistics for the Physical Sciences The smooth function f (x) is called a probability density function (pdf). Clearly f (x) must satisfy: f (x) ≥ 0; −∞ < x < 1; (1) Z 1 f (x) dx = 1: (2) −∞ Any f (x) satisfying these two conditions could be the pdf of some continuous random variable. 5 / 12 Continuous Random Variables Probability Density Function ST 380 Probability and Statistics for the Physical Sciences Uniform Distribution A bee leaves its hive to forage for blossom that will provide nectar. If the bee has no prior information, it searches in a random direction. If X is the direction, measured from North clockwise in degrees, then X is equally likely to be any value in [0; 360). More precisely, if 0 ≤ a < b < 360, then b − a Z b 1 P(a ≤ X ≤ b) = = dx: 360 a 360 So the pdf is f (x) = 1=360; 0 ≤ x < 360, and zero otherwise. 6 / 12 Continuous Random Variables Probability Density Function ST 380 Probability and Statistics for the Physical Sciences Cumulative Distribution Function The cumulative distribution function F (x) of any random variable X is defined as F (x) = P(X ≤ x); −∞ < x < 1: Earlier, for a continuous random variable X , Z b P(a ≤ X ≤ b) = f (y) dy; a so Z x F (x) = P(X ≤ x) = f (y) dy: −∞ 7 / 12 Continuous Random Variables Cumulative Distribution Function ST 380 Probability and Statistics for the Physical Sciences Conversely, dF (x) f (x) = = F 0(x): dx For the uniform distribution on [0; 1), ( 1 0 ≤ x < 1 f (x) = 0 otherwise and 8 0 x < 0 Z x <> F (x) = f (y) dy = x 0 ≤ x < 1 −∞ :>1 x ≥ 1 8 / 12 Continuous Random Variables Cumulative Distribution Function ST 380 Probability and Statistics for the Physical Sciences Percentiles The (100p)th percentile of X is the value that X falls below with probability p. That is, it is the value η(p) that satisfies P[X ≤ η(p)] = p: In terms of the cdf, Z η(p) p = F [η(p)] = f (y) dy: −∞ 9 / 12 Continuous Random Variables Cumulative Distribution Function ST 380 Probability and Statistics for the Physical Sciences Median and Quartiles Most commonly used percentiles: The median is the 50th percentile, 1 P(X ≤ median) = : 2 The upper and lower quartiles are the 75th and 25th percentiles, respectively, 3 P(X ≤ upper quartile) = ; 4 1 P(X ≤ lower quartile) = : 4 10 / 12 Continuous Random Variables Cumulative Distribution Function ST 380 Probability and Statistics for the Physical Sciences Expected Value Recall that the expected value of a discrete random variable is the average of its values, weighted by their probabilities. For a continuous random variable, expected value is defined the same way, but the average must be computed as an integral instead of a sum: Z 1 µX = E(X ) = x · f (x) dx: −∞ More generally, for a function h(X ), Z 1 µh(X ) = E[h(X )] = h(x) · f (x) dx: −∞ 11 / 12 Continuous Random Variables Expected Value ST 380 Probability and Statistics for the Physical Sciences Variance and Standard Deviation As for a discrete random variable, the variance of X is 2 2 σX = V (X ) = E (X − µX ) and the standard deviation is p σX = V (X ): But the expected value is now given by an integral: Z 1 2 2 E (X − µX ) = (x − µX ) · f (x) dx −∞ 12 / 12 Continuous Random Variables Expected Value.
Load more

Recommended publications
  • Fan Chart: Methodology and Its Application to Inflation Forecasting in India
    W P S (DEPR) : 5 / 2011 RBI WORKING PAPER SERIES Fan Chart: Methodology and its Application to Inflation Forecasting in India Nivedita Banerjee and Abhiman Das DEPARTMENT OF ECONOMIC AND POLICY RESEARCH MAY 2011 The Reserve Bank of India (RBI) introduced the RBI Working Papers series in May 2011. These papers present research in progress of the staff members of RBI and are disseminated to elicit comments and further debate. The views expressed in these papers are those of authors and not that of RBI. Comments and observations may please be forwarded to authors. Citation and use of such papers should take into account its provisional character. Copyright: Reserve Bank of India 2011 Fan Chart: Methodology and its Application to Inflation Forecasting in India Nivedita Banerjee 1 (Email) and Abhiman Das (Email) Abstract Ever since Bank of England first published its inflation forecasts in February 1996, the Fan Chart has been an integral part of inflation reports of central banks around the world. The Fan Chart is basically used to improve presentation: to focus attention on the whole of the forecast distribution, rather than on small changes to the central projection. However, forecast distribution is a technical concept originated from the statistical sampling distribution. In this paper, we have presented the technical details underlying the derivation of Fan Chart used in representing the uncertainty in inflation forecasts. The uncertainty occurs because of the inter-play of the macro-economic variables affecting inflation. The uncertainty in the macro-economic variables is based on their historical standard deviation of the forecast errors, but we also allow these to be subjectively adjusted.
    [Show full text]
  • Finding Basic Statistics Using Minitab 1. Put Your Data Values in One of the Columns of the Minitab Worksheet
    Finding Basic Statistics Using Minitab 1. Put your data values in one of the columns of the Minitab worksheet. 2. Add a variable name in the gray box just above the data values. 3. Click on “Stat”, then click on “Basic Statistics”, and then click on "Display Descriptive Statistics". 4. Choose the variable you want the basic statistics for click on “Select”. 5. Click on the “Statistics” box and then check the box next to each statistic you want to see, and uncheck the boxes next to those you do not what to see. 6. Click on “OK” in that window and click on “OK” in the next window. 7. The values of all the statistics you selected will appear in the Session window. Example (Navidi & Monk, Elementary Statistics, 2nd edition, #31 p. 143, 1st 4 columns): This data gives the number of tribal casinos in a sample of 16 states. 3 7 14 114 2 3 7 8 26 4 3 14 70 3 21 1 Open Minitab and enter the data under C1. The table below shows a portion of the entered data. ↓ C1 C2 Casinos 1 3 2 7 3 14 4 114 5 2 6 3 7 7 8 8 9 26 10 4 Now click on “Stat” and then choose “Basic Statistics” and “Display Descriptive Statistics”. Click in the box under “Variables:”, choose C1 from the window at left, and then click on the “Select” button. Next click on the “Statistics” button and choose which statistics you want to find. We will usually be interested in the following statistics: mean, standard error of the mean, standard deviation, minimum, maximum, range, first quartile, median, third quartile, interquartile range, mode, and skewness.
    [Show full text]
  • Binomial and Normal Distributions
    Probability distributions Introduction What is a probability? If I perform n experiments and a particular event occurs on r occasions, the “relative frequency” of r this event is simply . This is an experimental observation that gives us an estimate of the n probability – there will be some random variation above and below the actual probability. If I do a large number of experiments, the relative frequency gets closer to the probability. We define the probability as meaning the limit of the relative frequency as the number of experiments tends to infinity. Usually we can calculate the probability from theoretical considerations (number of beads in a bag, number of faces of a dice, tree diagram, any other kind of statistical model). What is a random variable? A random variable (r.v.) is the value that might be obtained from some kind of experiment or measurement process in which there is some random uncertainty. A discrete r.v. takes a finite number of possible values with distinct steps between them. A continuous r.v. takes an infinite number of values which vary smoothly. We talk not of the probability of getting a particular value but of the probability that a value lies between certain limits. Random variables are given names which start with a capital letter. An r.v. is a numerical value (eg. a head is not a value but the number of heads in 10 throws is an r.v.) Eg the score when throwing a dice (discrete); the air temperature at a random time and date (continuous); the age of a cat chosen at random.
    [Show full text]
  • On Median and Quartile Sets of Ordered Random Variables*
    ISSN 2590-9770 The Art of Discrete and Applied Mathematics 3 (2020) #P2.08 https://doi.org/10.26493/2590-9770.1326.9fd (Also available at http://adam-journal.eu) On median and quartile sets of ordered random variables* Iztok Banicˇ Faculty of Natural Sciences and Mathematics, University of Maribor, Koroskaˇ 160, SI-2000 Maribor, Slovenia, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia, and Andrej Marusiˇ cˇ Institute, University of Primorska, Muzejski trg 2, SI-6000 Koper, Slovenia Janez Zerovnikˇ Faculty of Mechanical Engineering, University of Ljubljana, Askerˇ cevaˇ 6, SI-1000 Ljubljana, Slovenia, and Institute of Mathematics, Physics and Mechanics, Jadranska 19, SI-1000 Ljubljana, Slovenia Received 30 November 2018, accepted 13 August 2019, published online 21 August 2020 Abstract We give new results about the set of all medians, the set of all first quartiles and the set of all third quartiles of a finite dataset. We also give new and interesting results about rela- tionships between these sets. We also use these results to provide an elementary correctness proof of the Langford’s doubling method. Keywords: Statistics, probability, median, first quartile, third quartile, median set, first quartile set, third quartile set. Math. Subj. Class. (2020): 62-07, 60E05, 60-08, 60A05, 62A01 1 Introduction Quantiles play a fundamental role in statistics: they are the critical values used in hypoth- esis testing and interval estimation. Often they are the characteristics of distributions we usually wish to estimate. The use of quantiles as primary measure of performance has *This work was supported in part by the Slovenian Research Agency (grants J1-8155, N1-0071, P2-0248, and J1-1693.) E-mail addresses: [email protected] (Iztok Banic),ˇ [email protected] (Janez Zerovnik)ˇ cb This work is licensed under https://creativecommons.org/licenses/by/4.0/ 2 Art Discrete Appl.
    [Show full text]
  • The Upper Quartile Point: Below). the Interquartile Range (IQR)
    502 Median, quartiles, and percentiles are statistical Sample Application (§) ways to summarize the location and spread of experi- Suppose the mental data. They are a robust form of data reduc- EF0-EF1 tornadoes) are: , where hundreds or thousands of data are rep- tion 1.5, 0.8, 1.4, 1.8, 8.2, 1.0, 0.7, 0.5, 1.2 resented by several summary statistics. Find the median and interquartile range. Compare First, your data from the smallest to largest sort with the mean and standard deviation. values. This is easy to do on a computer. Each data point now has a rank associated with it, such as 1st Find the Answer: (smallest value), 2nd, 3rd, ... nth (largest value). Let x Given: data set listed above. = the value of the rth Find: q r = (1/2)·(n+1)] 0.5 Mean σ is called the median, and the data value x of this middle data point is the median value (q ). Namely, 0.5 First sort the data in ascending order: q = x for n = odd 0.5 Values ( ): 0.5, 0.7, 0.8, 1.0, 1.2, 1.4, 1.5, 1.8, 8.2 If n is an even number, there is no data point exactly in r): 1 2 3 4 5 6 7 8 9 the middle, so use the average of the 2 closest points: Middle: ^ q = 0.5· [x + x ] for n = even 0.5 +1 Thus, the median point is the 5th The median is a measure of the location or center data set, and corresponding value of that data point is of the data median = q = z = 1.2 km.
    [Show full text]
  • Skewness and Kurtosis 2
    A distribution is said to be 'skewed' when the mean and the median fall at different points in the distribution, and the balance (or centre of gravity) is shifted to one side or the other-to left or right. Measures of skewness tell us the direction and the extent of Skewness. In symmetrical distribution the mean, median and mode are identical. The more the mean moves away from the mode, the larger the asymmetry or skewness A frequency distribution is said to be symmetrical if the frequencies are equally distributed on both the sides of central value. A symmetrical distribution may be either bell – shaped or U shaped. 1- Bell – shaped or unimodel Symmetrical Distribution A symmetrical distribution is bell – shaped if the frequencies are first steadily rise and then steadily fall. There is only one mode and the values of mean, median and mode are equal. Mean = Median = Mode A frequency distribution is said to be skewed if the frequencies are not equally distributed on both the sides of the central value. A skewed distribution may be • Positively Skewed • Negatively Skewed • ‘L’ shaped positively skewed • ‘J’ shaped negatively skewed Positively skewed Mean ˃ Median ˃ Mode Negatively skewed Mean ˂ Median ˂ Mode ‘L’ Shaped Positively skewed Mean ˂ Mode Mean ˂ Median ‘J’ Shaped Negatively Skewed Mean ˃ Mode Mean ˃ Median In order to ascertain whether a distribution is skewed or not the following tests may be applied. Skewness is present if: The values of mean, median and mode do not coincide. When the data are plotted on a graph they do not give the normal bell shaped form i.e.
    [Show full text]
  • Lecture Notes 5)
    Chapter 3 Continuous Random Variables 3.1 Introduction Rather than summing probabilities related to discrete random variables, here for continuous random variables, the density curve is integrated to determine probability. Exercise 3.1 (Introduction) Patient's number of visits, X, and duration of visit, Y . density, pmf f(x) density, pdf f(y) = y/6, 2 < y <_ 4 probability less than 1.5 = 1 sum of probability 1 at specific values probability less than 3 0.75 P(X < 1.5) = P(X = 0) + P(X = 1) 2/3 = 0.25 + 0.50 = 0.75 = area under curve, 0.50 P(Y < 3) = 5/12 1/2 probability at 3, P(X = 2) = 0.25 P(Y = 3) = 0 0.25 1/3 0 1 2 0 1 2 3 4 x probability (distribution): cdf F(x) 1 1 probability = 0.75 0.75 value of function, F(3) = P(Y < 3) = 5/12 0.50 probability less than 1.5 = value of function 0.25 F(1.5 ) = P(X < 1.5) = 0.75 0.25 0 1 2 0 1 2 3 4 x Figure 3.1: Comparing discrete and continuous distributions 73 74 Chapter 3. Continuous Random Variables (LECTURE NOTES 5) 1. Number of visits, X is a (i) discrete (ii) continuous random variable, and duration of visit, Y is a (i) discrete (ii) continuous random variable. 2. Discrete (a) P (X = 2) = (i) 0 (ii) 0:25 (iii) 0:50 (iv) 0:75 (b) P (X ≤ 1:5) = P (X ≤ 1) = F (1) = 0:25 + 0:50 = 0:75 requires (i) summation (ii) integration and is a value of a (i) probability mass function (ii) cumulative distribution function which is a (i) stepwise (ii) smooth increasing function (c) E(X) = (i) P xf(x) (ii) R xf(x) dx (d) V ar(X) = (i) E(X2) − µ2 (ii) E(Y 2) − µ2 (e) M(t) = (i) E etX (ii) E etY (f) Examples of discrete densities (distributions) include (choose one or more) (i) uniform (ii) geometric (iii) hypergeometric (iv) binomial (Bernoulli) (v) Poisson 3.
    [Show full text]
  • Teaching Quartile Location-Using Sample Size Divisibility Jon-Paul Paolino, Mercy College
    Teaching Quartile Location-Using Sample Size Divisibility Jon-Paul Paolino, Mercy College Quartiles are descriptive measures of location that can be introduced to students as early as primary school and are taught at the tertiary education level across the world. To successfully locate the quartiles of a univariate data set, basic counting and arithmetic are required. However, particularly in the tertiary-level statistics course, there is often confusion among students when textbooks and technology explain quartile location using complex computational algorithms (i.e., interpolation and cumulative distribution functions). We can eliminate such confusion by providing a table of closed-form solutions for locating the three quartiles. Review of Quartiles and Their Meanings Quartiles are the boundaries that divide a univariate data set into four (almost) equal subsets. These measures of location are instrumental in teaching topics such as interquartile range, boxplots, and Tukey’s outlier detection method. There exists a plethora of mathematically valid methods to estimate the quartiles, which run the gamut in difficulty from basic to rigorous. Computer software (i.e., Excel, SPSS, SAS, R, and STATA) is often incorporated, particularly at the tertiary level, for quick calculation. However, these technological resources are many times complicated to understand, because they give estimates based on highly sophisticated algorithms. Some textbooks are also at fault for basing their description of quartile location on these methods. This may leave students confused, especially in their first statistics class. First Quartile The first quartile, sometimes referred to as Q1, is a descriptive measure that indicates the 25th percentile of a univariate data set.
    [Show full text]
  • A Simple Glossary of Statistics. Barchart: Box-Plot: (Also Known As
    1 A simple glossary of statistics. Barchart: 3.5 Similar to a Histogram but the bars don‟t touch each other and the x-axis usually does not have a 3.0 continuous scale. 2.5 The example shows a bar chart of the colour of car 2.0 owned by ten people. (Note how the false origin 1.5 gives a first impression of even less blue cars.) 1.0 Count .5 red green black w hite blue Box-plot: (also known as box CARCOLOR and whisker plot) A Boxplot divides the data into quarters. The middle line shows the median (the value that Boxplot divides the data in half), the box shows the 110 range of the two middle quarters, and the 100 whisker whiskers show the range of the rest of the data. 90 upper quartile The values at the ends of the box are called the t h th g i 80 e quartiles, (SPSS refers to these as the 25 and W box th 70 m edian 75 percentiles) The distance between them is lower quartile 60 called the interquartile range (IQR). whisker 50 The more sophisticated version (which SPSS uses) marks outliers with circles, counting anything more than one and a half times the interquartile range away from the quartiles as an outlier, those over three times the interquartile range away from the quartiles are called extremes and marked with asterisks. The length of the box is equal to the interquartile range (IQR). Boxplots are most often used for comparing two or more sets of data.
    [Show full text]
  • Normal Curves Today’S Goals
    Normal Curves Today’s Goals • Normal curves! • Before this we need a basic review of statistical terms. I mean basic as in underlying, not easy. • We will learn how to retrieve statistical data from normal curves. • As an application, we’ll see how to determine the margin of error of a poll. Statistics basics Here’s some terminology you should be familiar with: • Mean/Average: For a set of N numbers, d1, d2,..., dN , the mean is given by µ = (d1 + d2 + ··· + dN )/N. • Median: Sort the data set from smallest to largest: d1, d2,..., dN . The median is the middle number. If N is odd, the median is d(N+1)/2. If N is even, the median is the average of dN/2 and d(N/2)+1. • Mode: The most common number(s). A data set can have more than one mode. (We won’t really study mode. It was just feeling left out so I put it on the slide.) • Range: The difference between the highest and lowest values of the data (R = Max − Min). Percentiles The pth percentiles of a data set is a number Xp such that p% is smaller or equal to Xp and (100 − p)% of the data is bigger or equal to Xp. To find the pth percentile of a sorted data set d1, d2,..., dN , first find the locator L = (p/100) N. dL+dL+1 If L is a whole number, then Xp = 2 . + If L is not a whole number, then Xp = dL+ where L is L rounded up.
    [Show full text]
  • Box-And-Whisker Plots 10.4
    10.4 Box-and-Whisker Plots How can you use quartiles to represent data graphically? 1 ACTIVITY: Drawing a Box-and-Whisker Plot Work with a partner. Numbers of Pairs of Footwear The numbers of pairs of footwear 2 5 12 3 owned by each student in a sixth grade 7 2 4 6 class are shown. 14 10 6 28 5 3 2 4 9 25 4 10 8 15 5 8 a. Order the data set from least to greatest. Then write the data on a strip of grid paper with 24 boxes. b. Use the strip of grid paper to fi nd the median, the fi rst quartile, and the third quartile. Identify the least value and the greatest value in the data set. c. Graph the fi ve numbers that you found in part (b) on the number line below. 0 5 10 15 20 25 30 35 d. The data display shown below is called a box-and-whisker plot. Fill in the missing labels and numbers. Explain how a box-and-whisker plot uses quartiles to represent the data. Data Displays In this lesson, you will ● make and interpret box-and-whisker plots. Pairs of footwear ● compare box-and-whisker 0 5 10 15 20 25 30 35 plots. e. Using only the box-and-whisker plot, which measure(s) of center can you fi nd for the data set? Which measure(s) of variation can you fi nd for the data set? Explain your reasoning. f. Why do you think this type of data display is called a box-and-whisker plot? Explain.
    [Show full text]
  • Mean Absolute Deviation Glossary What Is Statistics? Teacher Notes
    6th Grade Statistical Variability 2017­02­16 www.njctl.org Table of Contents Click on a topic to go to that section What is Statistics? Measures of Center Mean Median Mode Central Tendency Application Problems Measures of Variation Minimum/Maximum Teacher Notes Range Quartiles Outliers Mean Absolute Deviation Glossary What is Statistics? Teacher Notes Return to Table of Contents Data Everyday we encounter numbers. We use numbers throughout the day without even realizing it. What time did each of us wake up? How many minutes did we have to get ready? How far from school do we live? How many students will be in class today? How long is each class? And so on... Data This information comes at us in the form of numbers, and this information is called data. If we start trying to put together all of this data and make sense of it, we can get overwhelmed. Statistics is what helps us along. Statistics is the study of data. Data Conclusions MP7 ­ Look for and make use Lets put statistics to work! of structure. Two 6th grade math classes Ask: took the same end of unit test. • What do you notice about every data point within the Here are their results. same class? Class 1: • How do the two class' data Math Practice 89, 88, 92, 78, 85, 89, 95, 71, 100, 88, points relate to each other? 97, 82, 77, 98, 86, 82, 95, 88 • What patterns do you notice between the data points? Class 2: 100, 53, 92, 91, 97, 93, 92 What conclusions can you draw by looking at the numbers? Statistics When we just look at a list of numbers, it can take a lot of time to make sense of what we are dealing with.
    [Show full text]