Lecture_7_Sep_30_2010

Lecture_7_Sep_30_2010 - CONTINUOUS RANDOM VARIABLES AND...

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Unformatted text preview: CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS (CHAPTER 4 ) (CHAPTER Continuous Random Variables and Probability Distributions: Example Continuous Continuous Random Variables and Probability Distributions: Example Continuous Continuous Random Variables and Probability Distributions: Example Continuous Continuous Random Variables and Probability Distributions: Properties Continuous Continuous Random Variables and Probability Distributions: Example Continuous Continuous Random Variables and Probability Distributions: Properties Continuous Continuous Random Variables and Probability Distributions: Example Continuous Continuous Random Variables and Probability Distributions: Example Continuous Continuous Random Variables and Probability Distributions: Example Continuous Continuous Random Variables and Probability Distributions: Example Continuous Continuous Random Variables and Probability Distributions: Example Continuous Continuous Random Variables and Probability Distributions: Example Continuous Continuous Random Variables and Probability Distributions: Example Continuous Continuous Random Variables and Probability Distributions: Example Continuous Uniform Distribution Uniform The general formula for the probability density function of the uniform distribution is where A is the location parameter and (B - A) is the scale parameter. The case where A = 0 and B = 1 is called the standard uniform distribution. The equation for the standard uniform distribution is Uniform Distribution Uniform Uniform Distribution Uniform The formula for the cumulative distribution function of the uniform distribution is Exponential Distribution Exponential Exponential Distribution-Mean and Variance Exponential Exponential Distribution-Cumulative Distribution Function (cdf) Exponential Exponential Distribution-Plot of the Cumulative Distribution Function (cdf) Exponential Exponential Distribution-Example Exponential Standard Normal Distribution Standard Given a normal rv X,the general formula for the probability density function of the normal distribution is: normal where µ is the location parameter and σ is the scale parameter; and -∞ <x<∞ The case when µ =0 and σ =1 is called the standard normal distribution. The rv x standard (-∞ <x<∞ ) with the pdf f(x) shown below is called a standard normal rv. Standard normal distribution: The distribution is centred on 0, with 99.7% falling between −3 and 3 and 95% falling between −1.96 and 1.96. falling Standard Normal Distribution-cdf Standard Standard Normal Distribution Standard • If rv X follows the general normal distribution ie X~N(µ , σ 2) ; normal distribution then the rv Z obtained by the standardizing transformation: Z=(X―µ )/σ ; is a standard normal rv ie Z ∼ N(0, 1). standard • The cdf of Z, F(Z ≤ z) is typically denoted by φ (z) The • Tables for φ (z) can be found easily (Appendix Table A.3 in your text book). Various probabilities can be computed using these tables. • Because of the symmetry of this distribution, P(Z ≤ -z)= P(Z ≥ z) Standard Normal Distribution Standard Normal Distribution Normal • Every normal curve (regardless of its mean or standard deviation) conforms to the following "rule“: • About 68% of the area under the curve falls within 1 standard deviation of the mean. • About 95% of the area under the curve falls within 2 standard deviations of the mean. • About 99.7% of the area under the curve falls within 3 standard deviations of the mean. • Collectively, these points are known as the empirical rule or the 68-9599.7 rule. Clearly, given a normal distribution, most outcomes will be within 3 standard deviations of the mean. Normal Distribution Normal Normal Distribution Normal • • P( 0 ≤ Z ≤ 1 ) =0.34134 (approx half of 68%) P( 0 ≤ Z ≤ 1 ) = φ (1) - φ (0) (where φ (0) and φ (1) can be obtained from the standard normal tables in Appendix Tables A.3) Note φ (0) will always be 0.5, because, the std. normal rv is symmetric and centered around 0, so P(Z≤0)=P(Z≥ 0)=0.5. • ...
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