Normal Distribution. Definition A continuous rv X is said to have a normal distribution with. the pdf of X is

Normal Distribution

Normal Distribution Definition A continuous rv X is said to have a normal distribution with parameter µ and σ (µ and σ 2 ), where < µ < and σ > 0, if the pdf of X is f (x; µ, σ) = 1 2πσ e (x µ)2 /(2σ 2 ) We use the notation X N(µ, σ 2 ) to denote that X is rormally distributed with parameters µ and σ 2.

Normal Distribution Proposition For X N(µ, σ 2 ), we have E(X ) = µ and V (X ) = σ 2

Normal Distribution Definition The normal distribution with parameter values µ = 0 and σ = 1 is called the standard normal distribution. A random variable having a standard normal distribution is called a standard normal random variable and will be denoted by Z. The pdf of Z is f (z; 0, 1) = 1 2π e z2 /2 < z < The graph of f (z; 0, 1) is called the standard normal (or z) curve. The cdf of Z is P(Z z) = z f (y; 0, 1)dy, which we will denote by Φ(z).

Normal Distribution Proposition If X has a normal distribution with mean µ and stadard deviation σ, then Z = X µ σ has a standard normal distribution. Thus P(a X b) = P( a µ Z b µ σ σ ) = Φ( b µ σ ) Φ(a µ σ ) P(X a) = Φ( a µ µ ) P(X b) = 1 Φ(b σ σ )

Normal Distribution

Normal Distribution Proposition {(100p)th percentile for N(µ, σ 2 )} = µ + {(100p)th percentile for N(0, 1)} σ

Normal Distribution

Normal Distribution Proposition Let X be a binomial rv based on n trials with success probability p. Then if the binomial probability histogram is not too skewed, X has approximately a normal distribution with µ = np and σ = npq, where q = 1 p. In particular, for x = a posible value of X, ( ) area under the normal curve P(X x) = B(x; n, p) to the left of x+0.5 x+0.5 np = Φ( ) npq In practice, the approximation is adequate provided that both np 10 and nq 10, since there is then enough symmetry in the underlying binomial distribution.

Normal Distribution

Normal Distribution A graphical explanation for ( ) area under the normal curve P(X x) = B(x; n, p) to the left of x+0.5 x+0.5 np = Φ( ) npq

Normal Distribution

Normal Distribution Example (Problem 54) Suppose that 10% of all steel shafts produced by a certain process are nonconforming but can be reworked (rather than having to be scrapped). Consider a random sample of 200 shafts, and let X denote the number among these that are nonconforming and can be reworked. What is the (approximate) probability that X is between 15 and 25 (inclusive)? In this problem n = 200, p = 0.1 and q = 1 p = 0.9. Thus np = 20 > 10 and nq = 180 > 10

Exponential Distribution

Exponential Distribution Definition X is said to have an exponential distribution with parameter λ(λ > 0) if the pdf of X is { λe λx x 0 f (x; λ) = 0 otherwise

Exponential Distribution

Exponential Distribution Proposition If X EXP(λ), then And the cdf for X is E(X ) = 1 λ and V (X ) = 1 λ 2 F (x; λ) = { 1 e λx x 0 0 x < 0

Exponential Distribution

Exponential Distribution Memoryless Property for any positive t and t 0. P(X t) = P(X t + t 0 X t 0 )

Exponential Distribution Memoryless Property P(X t) = P(X t + t 0 X t 0 ) for any positive t and t 0. In words, the distribution of additional lifetime is exactly the same as the original distribution of lifetime, so at each point in time the component shows no effect of wear. In other words, the distribution of remaining lifetime is independent of current age.

Example: There is a machine available for cutting corks intended for use in wine bottles. We want to find out the distribution of the diameters of the corks produced by that machine. Assume we have 10 samples produced by that machine and the diameters is recorded as following: 3.0879 3.2546 2.8970 2.7377 2.7740 2.6030 3.5931 3.1253 2.4756 2.5133

3.0879 3.2546 2.8970 2.7377 2.7740 2.6030 3.5931 3.1253 2.4756 2.5133

Sample Percentile

Sample Percentile Recall: The (100p)th percentile of the distribution of a continuous rv X, denoted by η(p), is defined by p = F (η(p)) = η(p) f (y)dy.

Sample Percentile Recall: The (100p)th percentile of the distribution of a continuous rv X, denoted by η(p), is defined by p = F (η(p)) = η(p) f (y)dy. In words, the (100p)th percentile η(p) is the X value such that there are 100p% X values below η(p). Similarly, we can define sample percentile in the same manner, i.e. the (100p)th percentile x p is the value such that there are 100p% sample values below x p.

Definition Assume we have a sample with size n. Order the n sample observations from smallest to largest. Then the ith smallest observation in the list is taken to be the [100(i 0.5)/n]th sample percentile. Remark: 1. Why i 0.5? We regard the sample observation as being half in the lower group and half in the upper group. e.g. if n = 9, then the sample median is the 5th largest observation and this observation is regarded as two parts: one in the lower half and one in the upper half.

Example: for the previous example, the [100(i 0.5)/n]th sample percentile is tabulated as following: 2.4756 2.5133 2.6030 100(1-.5)/10 = 5% 100(2-.5)/10 = 15% 100(3-.5)/10 = 25% 2.7377 2.7740 100(4-.5)/10 = 35% 100(5-.5)/10 = 45% 2.8970 3.0879 3.1253 100(6-.5)/10 = 55% 100(7-.5)/10 = 65% 100(8-.5)/10 = 75% 3.2546 3.5931 100(9-.5)/10 = 85% 100(10-.5)/10 = 95%

Idea for Quantile-Quantile Plot: 1. Determine the [100(i 0.5)/n]th sample percentile for a given sample.

Idea for Quantile-Quantile Plot: 1. Determine the [100(i 0.5)/n]th sample percentile for a given sample. 2. Find the corresponding [100(i 0.5)/n]th percentile from the population with the assumed distribution; for example, if the assumed distribution is standard normal, then find corresponding [100(i 0.5)/n]th percentile from the standard normal distribution. 3. Consider the (population percentile, sample percentile) pairs, i.e. ([100(i ) 0.5)/n]th percentile, ith smallest sample of the distribution observation

4. Each pair plotted as a point on a two-dimensional coordinate system should fall close to a 45 line. Substantial deviations of the plotted points from a 45 line cast doubt on the assumption that the distribution under consideration is the correct one. Probability Plot Idea for Quantile-Quantile Plot: 1. Determine the [100(i 0.5)/n]th sample percentile for a given sample. 2. Find the corresponding [100(i 0.5)/n]th percentile from the population with the assumed distribution; for example, if the assumed distribution is standard normal, then find corresponding [100(i 0.5)/n]th percentile from the standard normal distribution. 3. Consider the (population percentile, sample percentile) pairs, i.e. ([100(i ) 0.5)/n]th percentile, ith smallest sample of the distribution observation

Example 4.29: The value of a certain physical constant is known to an experimenter. The experimenter makes n = 10 independent measurements of this value using a particular measurement device and records the resulting measurement errors (error = observed value - true value). These observations appear in the following table. Percentage 5 15 25 35 45 Sample Observation -1.91-1.25-0.75-0.53 0.20 Percentage 55 65 75 85 95 Sample Observation 0.35 0.72 0.87 1.40 1.56

We first find the corresponding population distribution percentiles, in this case, the z percentiles: Percentage 5 15 25 35 45 Sample Observation -1.91-1.25-0.75-0.53 0.20 z percentile -1.645-1.037-0.675-0.385-0.126 Percentage 55 65 75 85 95 Sample Observation 0.35 0.72 0.87 1.40 1.56 z percentile 0.126 0.385 0.675 1.037 1.645

What about the first example? We are only interested in whether the ten sample observations come from a normal distribution.

What about the first example? We are only interested in whether the ten sample observations come from a normal distribution. Recall: {(100p)th percentile for N(µ, σ 2 )} = µ + {(100p)th percentile for N(0, 1)} σ If µ = 0, then the pairs (σ [z percentile], observation) fall on a 45 line, which has slope 1. Therefore the pairs ([z percentile], observation) fall on a line passing through (0,0) (i.e., one with y-intercept 0) but having slope σ rather than 1.

Normal Probability Plot A plot of the n pairs ([100(i 0.5)/n]th z percentile, ith smallest observation) on a two-dimensional coordinate system is called a normal probability plot. If the sample observations are in fact drawn from a normal distribution with mean value µ and standard deviation σ, the points should fall close to a straight line with slope σ and y-intercept µ. Thus a plot for which the points fall close to some straight line suggests that the assumption of a normal population distribution is plausible.

Percentage 5 15 25 35 45 Sample Observation 2.4756 2.5133 2.6030 2.7377 2.7740 z percentile -1.645-1.037-0.675-0.385-0.126 Percentage 55 65 75 85 95 Sample Observation 2.8970 3.0879 3.1253 3.2546 3.5931 z percentile 0.126 0.385 0.675 1.037 1.645

A nonnormal population distribution can often be placed in one of the following three categories: 1. It is symmetric and has lighter tails than does a normal distribution; that is, the density curve declines more rapidly out in the tails than does a normal curve. 2. It is symmetric and heavy-tailed compared to a normal distribution. 3. It is skewed.

Symmetric and light-tailed : e.g. Uniform distribution

Symmetric and heavy-tailed: e.g. Cauchy distribution with pdf f (x) = 1/[π(1 + x 2 )] for < x <

Skewed: e.g. lognormal distribution

Some guidances for probability plot for normal distributions (from the book Fitting Equations to Data (2nd ed.) Daniel, Cuthbert, and Fed Wood, Wiley, New York, 1980)

Some guidances for probability plot for normal distributions (from the book Fitting Equations to Data (2nd ed.) Daniel, Cuthbert, and Fed Wood, Wiley, New York, 1980) 1. For sample size smaller than 30, there is typically greater variation in the apperance of the probability plot. 2. Only for much larger sample sizes does a linear pattern generally predominate. Therefore, when a plot is based on a small sample size, only a very substantial departure from linearity should be taken as conclusive evidence of nonnorality.

Definition Consider a family of probability distributions involving two parameters, θ 1 and θ 2, and let F (x; θ 1, θ 2 ) denote the corresponding cdf s. The parameters θ 1 and θ 2 are said to be location and scale parameters, respectively, if F (x; θ 1, θ 2 ) is a function of (x θ 1 )/θ 2. e.g. 1. Normal distributions N(µ, σ): F (x; µ, σ) = Φ( x µ σ ).

For Weibull distribution: F (x; α, β) = 1 e (x/β)α, the parameter β is a scale parameter but α is NOT a location parameter. α is usually referred to as a shape parameter.

For Weibull distribution: F (x; α, β) = 1 e (x/β)α, the parameter β is a scale parameter but α is NOT a location parameter. α is usually referred to as a shape parameter. Fortunately, if X has a Weibull distribution with shape parameter α and scale parameter β, then the transformed variable ln(x ) has an extreme value distribution with location parameter θ 1 = ln(β) and scale parameter θ 2 = 1/α.

The gamma distribution also has a shape parameter α. However, there is no transformation h( ) such that h(x ) has a distribution that depends only on location and scale parameters.

The gamma distribution also has a shape parameter α. However, there is no transformation h( ) such that h(x ) has a distribution that depends only on location and scale parameters. Thus, before we construct a probability plot, we have to estimate the shape parameter from the sample data.