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North Carolina State University STAT 370: Probabilityy and Statistics for Engineers [Section 002] Announcements • HW 9 (continuous R.V., 10pt) due Friday Mar 30 @ 6PM • Midterm 2 (100 pts) on Apr 2 (Learning Objectives posted on course website; ignore H7-H11) • No class on Apr 4 (Enjoy the spring holiday!) Instructor: Hua Zhou Harrelson Hall 210 11:45AM-1:00PM, Mar 28, 2012 Plan Last time: Discrete RV, Binomial distribution (pmf, mean/variance) Today: • Remark on Quiz 4 (always read problem carefully) • Continuous random variable (pdf, cdf, mean/variance) Continuous Random Variables • A continuous random variable is a random variable that takes on an uncountable number of possible values. (It takes values in intervals). Eg: The speed off the next car that passes the state trooper (continuous, takes values greater than zero). • Can take on any value in a given interval – Examples: height, weight, length, mass, temperature, concentration 1 Probability density function (PDF) Remarks on PDF • The probability density function (pdf) f(x) defines the distribution of a continuous RV by giving probability over an interval: • f(x) can be greater than 1 but integral over any (a,b): can only be 0 ≤ P(a ≤ X ≤ b) ≤ 1; b P (a X b) f ( x ) dx a • Necessary and sufficient conditions for f(x) to be a PDF 1. f(x) has to be nonnegative: f(x)>=0; ( ) is 1: 2. Total area under the curve f(x) • Don’t need to worry about “ < “ or “ ≤ “ when we talk about probability of a continuous RV: a P( X a) P(a X a) f ( x)dx 0 a f ( x)dx 1 and area under the curve for a specific interval (a,b) is the probability for the interval. One key difference between continuous and discrete RV is that in the continuous case there is no probability at a point; positive probability is only given to intervals. For discrete RV ….. Example: Uniform distribution • Uniform distribution on interval [0,1] • PDF: f(x) = 1 if 0<=x<=1 f(x) = 0 otherwise • Verify it is a PDF. • Calculate P(1/2<=X<=3/4) Example • Consider X to be a continuous RV with pdf given by: f(x) = 2x, 0 ≤ x ≤ 1 = 0 otherwise i) Verify that f(x) is indeed a pdf 1) f(x) is nonnegative for 0 ≤ x ≤ 1 2) it integrates to 1 1 f ( x)dx d 2 xdx d x2 ii) Calculate P(1/4<=X<=2/3) 1 0 1 0 2/ 3 P(1/ 4 X 2/ 3) 2xdx 4/ 9 1/16 55/144 1/ 4 2 In class exercise Home exercise Let • Assume X ~ has the pdf given by x f ( x) x • • • • -1 x 0 0 x 1 Show this is a valid density function Compute P ( 1/ 4 X 1/ 8) Calculate the mean and variance of X; Find the cdf of X, F (t ) P( X t ) , for all t 3 2 x f (x) 2 0 Example • Assume X ~ has the pdf given by x f ( x) x F : (, ) [0,1] [0 1] is defined as F (t ) P ( X t ) t otherw ise a) Show that f(x) is a valid pdf for some RV X b) Compute P(0<X<1/2) c) Compute P(-1/2<X<1/2) d) Compute the CDF F(t)=P(X ≤ t) for all t and draw its graph. g p e) Compute E(X) and Var(X) Cumulative Distribution Function (CDF) • Consider a continuous RV, X with PDF given by f(x) . The cumulative distribution function F(t) -1 x 1 -1 x 0 0 x 1 f ( x ) dx Necessary and N d sufficient ffi i t conditions diti ffor F(t) to t be b CDF: CDF 1. F ( ) 0; F ( ) 1 2. 0 F ( t ) 1 3. Non-decreasing function • • • • Show this is a valid density function Compute P ( 1/ 4 X 1/ 8) Calculate the mean and variance of X; Find the cdf of X, F (t ) P( X t ) , for all t. Draw its graph. 3 Example Relationship between CDF and PDF Given two independent RVs: Y and Z, both are Uniform on [0,1]. Consider the RV, X=max(Y,Z), which is the maximum of Y and Z • Remark: We integrate f(x) to get F(x). The fundamental theorem of calculus says that if we differentiate F(x) we get f(x). f(x) What is the CDF of X? F (t ) 0 if t 0 =t 2 if 0 t 1 =1 if t 1 What is the PDF of X? Mean and variance of a continuous RV with pdf f(x) • Mean of a continuous RV: • For continuous RV X with pdf f(x)=2x if 0 ≤ x ≤ 1 =0 otherwise, E(X ) xf ( x )dx • Mean • Variance of a continuous RV Var ( X ) (x EX ) 2 f ( x)dx x f ( x)dx ( EX ) Var ( X ) EX 2 2 (EX ) 1 xf ( x)dx 2 x dx 2 / 3 2 • Variance 2 E( X ) 2 Example 0 1 x 2 f ( x )dx (2 / 3) 2 2 x 3dx 4 / 9 1 / 18 0 2 4 In class exercise Home exercise The pdf of a random variable X is given by x0 0 x 0 x 1 f ( x) 2 x 1 x 2 0 2 x Consider a random variable with cumulative distribution function x0 0 1/ 2 0 x 1 F(x) 1 x 2 3/ 4 1 x2 a) Graph the function b) Compute P ( X 1 .55 ) c) Compute E(X). • Find mean, variance, and CDF of X Home exercise Consider a random variable with cumulative distribution function 0 2 x / 2 F ( x) 2 2 x ( x / 2) 1 1 x0 0 x 1 1 x 2 x2 a) Graph the function b) Compute C t P ( X 1 .55 ) c) Compute E(X) and Var(X) 5