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note10 - f x = cx< x< 2 otherwise(f Find E X(g Find Var...

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Chapter 2 Section 4: Continuous Random Variables The distributions of continuous r.v.s can be specified using probability density function (pdf): f ( x ) cumulative distribution function (cdf) : F ( x ) pdf for a continuous random variable X is: a nonnegative function f ( x ) integraltext -∞ f ( x )d x = 1 For all a < b , P ( a X b ) = integraltext b a f ( x )d x . P ( X a ) = integraltext a -∞ f ( x )d x and P ( X b ) = integraltext b f ( x )d x cdf for continuous random variable X is: continuous non-decreasing function 0 F ( x ) 1: starts at 0 and ends at 1 F ( x ) = P ( X x ) = integraltext x f ( t )d t where f ( t ) is the pdf Note : (a) P ( a X b ) = P ( a < X b ) = P ( a X < b ) = P ( a < X < b ) (b) P ( X = a ) = integraltext a a f ( x )d x = 1
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Ex1 : Suppose that X has the following distribution. f ( x ) = cx 0 < x < 2 0 otherwise (a) Find the value of c . (b) Find the probability that x is between 0 and 1. (c) sketch a graph of the pdf f ( x ). (d) Find F ( x ). (e) Sketch a graph of the cdf F ( x ). 2
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Mean and Variance of Continuous Random Variable E( X ) = μ = integraltext -∞ xf ( x )d x Var( X ) = σ 2 = integraltext -∞ ( x - μ ) 2 f ( x )d x = integraltext -∞ x 2 f ( x )d
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Unformatted text preview: f ( x ) = cx < x < 2 otherwise (f) Find E( X ) (g) Find Var( X ) Percentiles of a continuous probability distribution: • The p th percentile of the distribution is the value x p such that p % of the population values are less than or equal to x p . (0 ≤ p ≤ 100) • If X is a continuous r.v. with pdf f ( x ), the p th percentile of X is the point x p which solves the equation F ( x p ) = P ( X ≤ x p ) = I x p-∞ f ( x )d x = p 100 . • Median is the 50th percentile ( p = 50). In the above equation, solve for x p by putting p = 50. 3 Ex2 : Suppose that X has the following distribution. f ( x ) = . 10 0 < x < 10 otherwise (a) Find E( X ). (b) Find Var( X ). (c) Find F ( x ). (d) Find the median. 4...
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