IT College Introduction to Computer Statistical Packages Eng. Heba Hamad 2009

Chapter 6 (part 1) Normal Probability Distribution

Overview A continuous random variable can assume any value in an interval on the real line or in a collection of intervals. It is not possible to talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the random variable assuming a value within a given interval.

Continuous Probability Distributions n The probability of the random variable assuming a value within some given interval from x 1 to x 2 is defined to be the area under the graph of the probability density function between x 1 and x 2. f ( x ) x Uniform x1 x1x1 x1 x2 x2x2 x2 x f ( x ) Normal x1 x1x1 x1 x2 x2x2 x2 x1 x1x1 x1 Exponential x f ( x ) x1 x1x1 x1 x2 x2x2 x2

Uniform Probability Distribution The uniform probability distribution is perhaps the simplest distribution for a continuous random variable. This distribution is rectangular in shape and is defined by minimum and maximum values ab P(x)

A random variable is uniformly distributed whenever the probability is proportional to the interval’s length. Uniform Probability Distribution Examples: Suppose the time to open the webpage of the university of Palestine is uniformly distributed with minimum time of 20 seconds and a maximum value of 60 seconds The time to fly via a commercial airline from Orlando, Folrida to Atlanta, Georgia ranges from 60 min to 120 min. The random variable is the flight time within this interval

where: a = smallest value the variable can assume b = largest value the variable can assume f ( x ) = 1/( b – a ) for a < x < b = 0 elsewhere The uniform probability density function is: Uniform Probability Distribution

Var( x ) = ( b - a ) 2 /12 E( x ) = ( a + b )/2 Uniform Probability Distribution Expected Value of x Variance of x

Using Area to Find Probability

Uniform Probability Distribution Example 1: Slater customers are charged for the amount of salad they take. Sampling suggests that the amount of salad taken is uniformly distributed between 5 ounces and 15 ounces.

n Uniform Probability Density Function f ( x ) = 1/10 for 5 < x < 15 = 0 elsewhere where: x = salad plate filling weight Uniform Probability Distribution

n Expected Value of x n Variance of x E( x ) = ( a + b )/2 = (5 + 15)/2 = 10 Var( x ) = ( b - a ) 2 /12 = (15 – 5) 2 /12 = 8.33 Uniform Probability Distribution

for Salad Plate Filling Weight f(x)f(x) x /10 Salad Weight Uniform Probability Distribution

f(x)f(x) x /10 Salad Weight P(12 < x < 15) = 1/10(3) =.3 What is the probability that a customer will take between 12 and 15 ounces of salad? 12 Uniform Probability Distribution

Southwest Arizona State University provides bus service to students while they are on campus. A bus arrives at the North Main Street every 30 min between 6 a.m. and 11 p.m. during weekdays. Students arrive at the bus stop at random times. The time that a student waits is uniformly distributed from 0 to 30 min Example 2

n Draw a graph of this distribution n Show that the area of this uniform distribution is 1.00 n How long will a student “typically” have to wait for a bus? In other words what is the mean waiting time? What is the standard deviation of the waiting times? n What is the probability a student will wait more than 25 minutes? n What is the probability a student will wait between 10 and 20 minutes? Uniform Probability Distribution

The Family of Normal Probability Distribution Next we consider the normal probability distribution. Unlike the uniform distribution, the normal probability distribution has a very complex formula

Normal Probability Distributions Heights of people n It has been used in a wide variety of applications: Scientific measurements

Amounts of rainfall n It has been used in a wide variety of applications: Test scores

The distribution is symmetric; its skewness measure is zero. Normal Probability Distribution n Characteristics x

The entire family of normal probability distributions is defined by its mean  and its standard deviation . Normal Probability Distribution n Characteristics Standard Deviation  Mean  x

The highest point on the normal curve is at the mean, which is also the median and mode. Normal Probability Distribution n Characteristics x

Normal Probability Distribution n Characteristics The mean can be any numerical value: negative, zero, or positive. x

Normal Probability Distribution n Characteristics  = 15  = 25 The standard deviation determines the width of the curve: larger values result in wider, flatter curves. x

Standard Normal Distribution This section presents the standard normal distribution which has three properties: 1. It is bell-shaped. 2. It has a mean equal to It has a standard deviation equal to 1. It is extremely important to develop the skill to find areas (or probabilities or relative frequencies) corresponding to various regions under the graph of the standard normal distribution.

Continuous Probability Distributions The graph of a discrete probability distribution is called a probability Histogram The graph of a continuous probability distribution is called a density curve

A density curve is the graph of a continuous probability distribution. It must Satisfy the following properties: Definition 1. The total area under the curve must equal Every point on the curve must have a vertical height that is 0 or greater. (That is, the curve cannot fall below the x-axis.)

A density curve for a uniform distribution is a horizontal line, so its easy to find the area of any underlying region by multiplying width and height. The density curve of a normal distribution has the bell shape. There are many different normal distributions with each one depending on two parameter ( the population mean and the population standard deviation).