Continuous Probability Distributions Uniform Probability Distribution Area as a measure of Probability The Normal Curve The Standard Normal Distribution Computing Probabilities for a Standard Normal Distribution f(x) X

Uniform Probability Distribution Chicago NY Consider the random variable x representing the flight time of an airplane traveling from Chicago to NY. Under normal conditions, flight time is between 120 and 140 minutes. Because flight time can be any value between 120 and 140 minutes, x is a continuous variable.

Uniform Probability Distribution With every one-minute interval being equally likely, the random variable x is said to have a uniform probability distribution

Uniform Probability Distribution For the flight-time random variable, a = 120 and b = 140

Uniform Probability Density Function for Flight time The shaded area indicates the probability the flight will arrive in the interval between 120 and 140 minutes

Basic Geometry Remember when we multiply a line segment times a line segment, we get an area

Probability as an Area Question: What is the probability that arrival time will be between 120 and 130 minutes—that is: 10

Notice that in the continuous case we do not talk of a random variable assuming a specific value. Rather, we talk of the probability that a random variable will assume a value within a given interval.

E(x) and Var(x) for the Uniform Continuous Distribution Applying these formulas to the example of flight times of Chicago to NY, we have: Thus

Normal Probability Distribution The normal distribution is by far the most important distribution for continuous random variables. It is widely used for making statistical inferences in both the natural and social sciences.

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

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

The Normal Distribution Where: μ is the mean σ is the standard deviation  = e =

The distribution is symmetric, and is bell-shaped. The distribution is symmetric, and is bell-shaped. Normal Probability Distribution n Characteristics x

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

The highest point on the normal curve is at the The highest point on the normal curve is at the mean, which is also the median and mode. mean, which is also the median and mode. The highest point on the normal curve is at the The highest point on the normal curve is at the mean, which is also the median and mode. 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, The mean can be any numerical value: negative, zero, or positive. zero, or positive. The mean can be any numerical value: negative, The mean can be any numerical value: negative, zero, or positive. 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. The standard deviation determines the width of the curve: larger values result in wider, flatter curves. x

Probabilities for the normal random variable are Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Probabilities for the normal random variable are Probabilities for the normal random variable are given by areas under the curve. The total area given by areas under the curve. The total area under the curve is 1 (.5 to the left of the mean and under the curve is 1 (.5 to the left of the mean and.5 to the right)..5 to the right). Normal Probability Distribution n Characteristics.5.5 x

The Standard Normal Distribution The Standard Normal Distribution is a normal distribution with the special properties that is mean is zero and its standard deviation is one.

 0 z The letter z is used to designate the standard The letter z is used to designate the standard normal random variable. normal random variable. The letter z is used to designate the standard The letter z is used to designate the standard normal random variable. normal random variable. Standard Normal Probability Distribution

Cumulative Probability 0 1 z Probability that z ≤ 1 is the area under the curve to the left of 1.

What is P(z ≤ 1)? Z ● ● ● ● ● To find out, use the Cumulative Probabilities Table for the Standard Normal Distribution

Exercise a)What is P(z ≤2.46)? b)What is P(z ≥2.46)? Answer: a).9931 b) =.0069 z

Exercise a)What is P(z ≤-1.29)? b)What is P(z ≥-1.29)? Answer: a) =.0985 b).9015 Note that, because of the symmetry, the area to the left of is the same as the area to the right of Red-shaded area is equal to green- shaded area Note that: z

Exercise 3 0 What is P(.00 ≤ z ≤1.00)? 1 P(.00 ≤ z ≤1.00)=.3413 z

Exercise 4 0 What is P(-1.67 ≥ z ≥ 1.00)? 1 P(-1.67 ≤ z ≤1.00)= Thus P(-1.67 ≥ z ≥ 1.00) = =.2062 z