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Lecture Note, July 14, 2014 Chih-Hsin Hsueh I. Binomial Distribution – Example It is known that screws produced by a certain company will be defective with probability .01 independently of each other. The company sells the screws in packages of 10 and offers a money-back guarantee that at most 1 of the 10 screws is defective. What proportion of packages sold must the company replace? II. Continuous Random Variable • Probability density function – The probability distribution of a continuous random variable is known as a continuous probability distribution. It can be described by a curve, called density curve. – Properties of density curve 1. The probability under density curve y = f (x) is equal to 1. 2. P (a ≤ X ≤ b) = area under curve between a and b. 3. F (a) = P (X ≤ a) is called the cumulative distribution function (cdf) of the random variable X, which is the area under the curve to the left of a. 4. 0 ≤ F (X) ≤ a and F (X) increases with a. • Mean and Variance for continuous random variable: Let X be a continuous random variable with probability distribution f (x). Then the mean (expectation/expected value), variance and standard deviation of X are given by the following : Z µ = E(X) = xf (x)dx Z 2 σ = V ar(X) = (x − µ)2 f (x)dx √ σ = Sd(X) = σ 2 1 • Some facts: III. Normal Distribution • It is a special type of density curve which describes a particular type of continuous probability distribution, called a normal distribution. These density curves are symmetric, unimodal(one peak) and bell-shaped. • It is one type of continuous probability distribution, which is described by a normal curve. It is a two- parameter distribution (the mean, µ, and the standard deviation, σ). This distribution is very much useful and has a wide application in the field of statistical inference. • Density function, mean, variance for Normal distribution: • Standard Normal distribution: A normal distribution with mean 0 and standard deviation 1 is called the standard normal distribution, denoted by N (0, 1) • Any normal random variable X with mean and standard deviation can be transformed into a random variable Z having a standard normal distribution via the standardization: z= 2 X −µ σ 3