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UCLA STATS 13 - Continuous Random Variables

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1STAT 13, UCLA, Ivo DinovSlide 1UCLA STAT 13Introduction toStatistical Methods for the Life and Health ScienceszInstructor: Ivo Dinov, Asst. Prof. In Statistics and NeurologyzTeaching Assistants: Janine Miller and Ming ZhengUCLA StatisticsUniversity of California, Los Angeles, Winter 2003http://www.stat.ucla.edu/~dinov/courses_students.htmlSTAT 13, UCLA, Ivo DinovSlide 2Chapter 6: Continuous Random VariableszContinuous Random VariableszThe Normal DistributionzSums and differences of random quantitiesSTAT 13, UCLA, Ivo DinovSlide 3200 400 600 800 Carbohydrate (mg/day)0.0020 600 800225 375.000.004(a) Standardized histogramShaded area = .483(Corresponds to 48.3% of observations)(b) Area between a = 225 and b = 375 shaded.002.000.004Dietary intake of carbohydrate (mg/day) for N=5929 people from a variety of work environments. Standardized histogram plot is unimodal but skewed to the right (high values). Vertical scale is (relative freq.)/(interval width) = fj/(N*m). The proportion of the data in [a : b] is the areaunder the standardized histogram on the range [a: b].mSTAT 13, UCLA, Ivo DinovSlide 4(d) Area between a = 225 and b = 375 shaded(c) With approximating curveShaded area = .486(cf. area = .483 for histogram).002.000.004.002.000.004Carbohydrate (mg/day)200 400 600 8000 0 600800225 375Superposition of a smooth curve (density function) on the standardized histogram (left panel). Area under the density curve on [a: b] = [225: 375] is analytically computed to be: 0.486 (right panel), which is close to the empirically obtained estimateof the area under the histogram on the same interval: 0.483 (left panel).STAT 13, UCLA, Ivo DinovSlide 5For a standardizedhistogram:z The vertical scale isRelative_frequency / Interval_widthz Total area under histogram = 1z Proportion of the data between a and bis the area under histogram between a and b.002.000.004Carbohydrate (mg/day)200 400 6000Standardized histogramsSTAT 13, UCLA, Ivo DinovSlide 6Probability and areasFor a continuous Xz the probability a random observation falls between aand b = area under the density curve between a and b.(d) Area between a = 225 and b = 375 shadedShaded area = .486(cf. area = .483 for histogram).002.000.0040600800225 3752STAT 13, UCLA, Ivo DinovSlide 7100 200 300 400 500 600100 200 300 400 500 600(a) Dot plots of 6 sets of 15 random observationsSampling from the distribution in Fig. 6.1.16 sets ofsamples of15 obs’sNote the fair amount of intra- and inter-group variability. What does that mean?Is that normal or expected?STAT 13, UCLA, Ivo DinovSlide 9100 200 300 400 500 600100 200 300 400 500 600100 200 300 400 500 600500,000 observations5 million observations100 200 300 400 500 600100 200 300 400 500 6005000 observations50,000 observations(b) Histograms with density curve superimposedFrom Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.STAT 13, UCLA, Ivo DinovSlide 10XVisualizing the population meanThe population mean is the imaginary value of X where the density curve balancesSTAT 13, UCLA, Ivo DinovSlide 11Recall a continuous variable is one where the domain has no gaps in between the values the variable can take.In calculations involving a continuous random variablewe do not have to worry about whether interval endpoints are included or excluded.Interval endpoints and continuous variablesSTAT 13, UCLA, Ivo DinovSlide 12Reviewz How does a standardized histogram differ from a relative-frequency histogram? raw histogram? (fj/mn)z What graphic feature conveys the proportion of the datafalling into a class interval for a standardized histogram? for a relative-frequency histogram? (area=width . height = m fj/mn= fj/n)z What are the two fundamental ways in which random observations arise? (Natural phenomena, sampling experiments – choose a student at random and use the lottery method to record characteristics, scientific experiments - blood pressure measure)z How does a density curve describe probabilities? (The probability that a random obs. falls in [a:b] is the area under the PDF on the same interval.)STAT 13, UCLA, Ivo DinovSlide 13Reviewz What is the total area under both a standardized histogram and a probability density curve? (1)z When can histograms of data from a random process be relied on to closely resemblethe density curve for that process? (large sample size, small histogram bin-size)z What characteristic of the density curve does the mean correspond to? (imaginary value of X, where the density curve balances)3STAT 13, UCLA, Ivo DinovSlide 14Reviewz Does it matter whether interval endpoints are included or excluded when we calculate probabilities for a continuous random variable from the area? (No)z Why? (Area[a:b] == Area(a:b))z Are discrete variables the same or different in this regard, interval endpoint not effecting the area? (Different)STAT 13, UCLA, Ivo DinovSlide 154035 450.00.10.2150 160 170 180 190 200.00.02.04.06(a) Chest measurements of Quetelet’s Scottish soldiers (in.)(b) Heights of the 4294 men in the workforce database (cm)= 39.8 in., = 2.05 in.= 174 cm, = 6.57 cmNormal density curve hasNormal density curve hasTwo standardized histograms with approximating Normal density curveSTAT 13, UCLA, Ivo DinovSlide 16z Is symmetric about the mean! Bell-shaped and unimodal.z Mean = Median!50% 50%Mean2.2The Normal distribution density curveN(µ, σµ, σµ, σµ, σ)STAT 13, UCLA, Ivo DinovSlide 17Effects of µµµµand σσσσ140 160 180shifts the curve along the axis2002=1742= 61=(a) Changing1= 160160 180 2001401= 62= 122=1701=increases the spread and flattens the curve(b) IncreasingMean is a measure of …central tendencyStandard deviation is a measure of …variability/spreadSTAT 13, UCLA, Ivo DinovSlide 18Understanding the standard deviation: σσσσ(c) Probabilities and numbers of standard deviationsShaded area = 0.683 Shaded area = 0.954 Shaded area = 0.997 68% chance of fallingbetween and− ++ 95% chance of fallingbetween and+2+23+ 99.7% chance of fallingbetween and3+−2−3−3−−2Probabilities/areas and numbers of standard deviationsfor the Normal distributionSTAT 13, UCLA, Ivo DinovSlide 19xArea = pr (X x )xArea = pr (X x )ORProbabilities supplied by computer programs –Cumulative (lower-tail) probabilitiesAreas in [0:Z] of the Std.Normal DistributionAre obtained by STATA command ztable4STAT 13, UCLA, Ivo DinovSlide 20Probabilities supplied by computer


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UCLA STATS 13 - Continuous Random Variables

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