Chapter 5 and 6 - Revised on Chapters 5 and 6 Some Commonly...

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Revised on November 12, 2011 Chapters 5 and 6 Some Commonly Used Distributions Sections 5.1, 5.2, 6.1 and 6.2 Random Variables Definition: A random variable is a function that assigns numerical value to each outcome in a sample space. Random variables are usually denoted by capital letters that are at the end of the alphabet. Quantitative random variables can be either discrete or continuous. A random variable is said to be discrete if its values can be listed. A continuous random variable has values within an interval of the real line, or a union of intervals or the whole real line. Comparison of the discrete and continuous random variables Properties Discrete Continuous Possible values Can be listed Infinitely many Probability distribution p(x) = P(X=x) (pmf) f(x) (pdf) Conditions for function 0 ≤ p(x) ≤ 1 & ( ) 1 all x p x = f(x) ≥ 0 & ( ) 1 f x dx -∞ = Cum. Dist. function (cdf) ( ) ( ) ( ) t x F x P X x p t = = . ( ) ( ) ( ) x F x P X x f t dt -∞ = = . E(X) = Mean (μ) ( ) X all x xp x μ = ( ) X xf x dx μ -∞ = Variance (σ 2 ) 2 2 ( ) ( ) X X all x x p x σ μ = - 2 2 ( ) ( ) X X x f x dx σ μ -∞ = - Standard Deviation (σ) 2 X X σ σ = + 2 X X σ σ = + Properties of the cdf: 1. Non-decreasing with 0 ≤ F(x) ≤ 1. 2. lim ( ) 0 lim ( ) 1. x x F x and F x →-∞ →+∞ = = 3. Discrete rvs have F(x) that is a step-function. 4. Continuous, except at, at most a finite number of points STA3032, Chapters 5 & 6, Page 1 of 8
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Empirical Rule: For a mound shaped, somewhat symmetric distribution of a random variable X, with mean μ X and standard deviation σ X , ( 29 ( 29 ( 29 1 0.68 2 0.95 3 1.00 X X X X X X P X P X P X μ σ μ σ μ σ - < × - < × - < × Chebyshev’s Inequality Let X be a random variable with mean μ and standard deviation σ. Then for any k > 1, ( 29 2 1 1 X X P X k k μ σ - - . Equivalently ( 29 2 1 X X P X k k μ σ - Sections 5.3 to 5.7 The Binomial Family of distributions The binomial family includes the Bernoulli distribution, Binomial distribution, Geometric distribution, Negative Binomial distribution, Poisson distribution and hypergeometric distribution. In all but the last one, we assume that there is an experiment with two possible outcomes, labeled “Success” (or S = the outcome we are interested in and count) and “Failure” (or F) You should notice that these distributions are all inter-related. There are similarities and differences between them. The differences define when you should use each one. Here is a list of these distributions with their similarities and differences: Distributions and Conditions when they are used: In each of the following distributions, we are assuming that a Bernoulli experiment (with only two possible outcomes, S and F) is repeated.
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