Normal Distribution

Normal Distribution - Stat 48 Lecture May 9th Normal...

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Stat 48 Lecture May 9 th Normal Distribution Dr Linda Penas Spring 2007 1 Continuous Distributions (p. 106) DEFN : A random variable is said to be continuous if its values can be put into a 1-1 correspondence to the real numbers. (i.e., measured on a continuous scale) Continuous Prob. Distributions DEFN : probability density function (pdf): for a continuous random variable X is a curve such that the area under the curve between two points a and b is equal to the probability that the random variable X falls between a and b. Continuous Prob. Distributions Cumulative Distribution Function (cdf ) of a continuous random variable X is given by () ( ) x Fx PX x ftd t −∞ =≤ = P(a < X < b) = area under the curve between a and b = F(b) - F(a) ab P(a < X < b) pdf must be nonnegative and the entire area under the curve is 1. For continuous random variables only, = ( ) ( ) P aXb P aXb P P ≤≤= <≤ ≤< =< < Q: Why does that work for continuous rvs? A: Because the area of a single point = 0 P(X = a) = 0 ( ) 0 a a Pa X a f xdx ≤= =
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Stat 48 Lecture May 9 th Normal Distribution Dr Linda Penas Spring 2007 2 NORMAL DISTRIBUTION The normal or Gaussian distribution is the cornerstone of most methods of estimation and hypothesis. NORMAL DISTRIBUTION Can be used to model things such as birth weight, blood pressures, lifelengths of components, etc Even random variables which are not normally distributed may often be approximated by the normal distribution. Defn : A continuous random variable X is said to have a normal distribution with mean μ and variance, σ 2 , if X has pdf given by: () 2 2 2 2 1 (; , ) e x p , , 2 2 , 0 x nx x μ μσ σ πσ ⎧⎫ ⎪⎪ =−− < < ⎨⎬ ⎩⎭ −∞ < < ∞ > We write 2 ~ ( , ) XN A distribution with pdf, f(x), is said to be symmetric about c if you can draw a vertical line passing through c, cut the curve in half using that vertical line, flip one side over and the two sides coincide for all x.
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This note was uploaded on 04/10/2008 for the course STAT 48 taught by Professor Penas during the Spring '08 term at UC Riverside.

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Normal Distribution - Stat 48 Lecture May 9th Normal...

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