Normal Distribution. Notes. Normal Distribution. Standard Normal. Sums of Normal Random Variables. Normal. approximation of Binomial.

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1 Lecture 21,22, 23 Text: A Course in Probability by Weiss 8.5 STAT 225 Introduction to Probability Models March 31, 2014 Standard Sums of Whitney Huang Purdue University 21,22, 23.1 Agenda 1 2 Standard 3 Sums of Standard Sums of 4 21,22, 23.2

2 Characteristics of the random variable: Let X be a r.v. The support for X: (, ) Its parameter(s) and definition(s): µ : mean and σ 2 : variance 1 The probability density function (pdf): e (x µ)2 2σ 2 2πσ 2 for < x < The cumulative distribution function (cdf): No explicit form, look at the value Φ( x µ σ ) for < x < from standard normal table The expected value: E[X] = µ The variance: Var(X) = σ 2 Standard Sums of 21,22, 23.3 Probability density function for X~N(0,1) Y~N(2,0.64) Z~N(-0.5,1.44) f(x) X Y Z Standard Sums of x The parameter µ determines the center of the distribution The parameter σ 2 determines the spread of the distribution Also called bell-shaped distribution 21,22, 23.4

3 Standard Z N(µ = 0, σ 2 = 1) random variable X with mean µ and standard deviation σ can convert to standard normal Z by the following : Z = X µ σ The cdf of the standard normal, denoted by Φ(z), can be found from the standard normal table The probability P(a X b) where X N(µ, σ 2 ) can be compute P(a X b) = P( a µ σ = Φ( b µ σ ) Φ(a µ σ ) Z b µ σ ) Standard Sums of 21,22, 23.5 Properties of Φ Φ(0) =.50 Mean and Median (50 th percentile) for standard normal are both 0 Φ( z) = 1 Φ(z) P(Z > z) = 1 Φ(z) = Φ( z) Standard Sums of 21,22, 23.6

4 Example 53 Let us examine Z. Find the following probabilities with respect to Z : 1 Z is at most Z is between 2 and 2 inclusive 3 Z is less than.5 Standard Sums of 21,22, 23.7 Example 53 cont d Solution. Standard Sums of 21,22, 23.8

5 Example 54 Find the following percentile with respect to Z 1 10 th percentile 2 55 th percentile 3 The third quartile 4 90 th percentile Standard Sums of 21,22, 23.9 Example 54 cont d Solution. Standard Sums of 21,22, 23.10

6 Example 55 Let X be with a mean of 20 and a variance of 49. Find the following probabilities and percentile: 1 X is between 15 and 23 2 X is more than 30 3 X is more than 12 knowing it is less than 20 4 What is the value that is smaller than 20% of the distribution? Standard Sums of 21,22, Example 55 cont d Solution. Standard Sums of 21,22, 23.12

7 Sums of If X i 1 i n are independent normal random variables with mean µ i are variance σi 2, respectively. Let S n = n i=1 X i then S n N( n i=1 µ i, n i=1 σ2 i ) This can be applied for any integer n Standard Sums of 21,22, Example 56 Let X 1, X 2, and X 3 be mutually independent, random variables. Let their means and standard deviations be 3k and k for k = 1, 2, and 3 respectively. Find the following distributions: 1 3 i=1 X i 2 X 1 + 2X 2 3X 3 3 X 1 + 5X 3 Standard Sums of 21,22, 23.14

8 Example 56 cont d Solution. Standard Sums of 21,22, We can use a to approximate a if n is large and p is close to.5 Rule of thumb for this approximation to be valid (in this class) is np > 5 and n(1 p) > 5 If X (n, p) with np > 5 and n(1 p) > 5 then we can use X N(µ = np, σ 2 = np(1 p)) to approximate X Notice that is a discrete distribution but normal is a continuous distribution so that P(X = x) = 0 x By using continuity correction we use P(x 0.5 X x + 0.5) to approximate P(X = x) Standard Sums of 21,22, 23.16