Continuous Probability Distributions

Discrete vs. Continuous

•Discrete▫A random variable (RV) that can take only

certain values along an interval: Cars passing by a point Results of coin toss Students taking a class

•Continuous▫An RV that can take on any value at any

point along an interval.

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Continuous Probability Distributions

•Discrete: For any random variable X: P(X=x)

•Continuous: ▫The probability that a continuous random

variable will assume a specific value is zero▫Therefore, a continuous random variable

cannot be expressed in tabular form.▫An equation or formula is used to describe

a continuous random variable. This is called a probability density function (pdf)

Limits (kind of)

 

•The random variable is a function of X▫y = f(x)

•The value of f(x) is greater than or equal to zero for all values of x.

•The total area under the curve always equals one.

Probability Density Functions

Continuous Probability Distributions

Let’s assume that a train arrives at the station precisely every 30 minutes.

If passengers arrive at the station at random intervals, what is the probability…?

Continuous Distributions

•Normal distribution•Standard normal distribution•Exponential distribution•Chi-square distribution•F distribution

Normal Distribution

•Carl Friedrich Gauss

Normal Distribution

•Many natural and economic phenomena are normally distributed

•The normal can approximate other distributions, including the binomial

•Sample proportions are normally distributed when taken from a population of any distribution

•Normal is a family of distributions▫Mean, median, and mode all at the same

position▫Curve is symmetric▫Curve is asymptotic

pdf for the Normal

 

2σ

Empirical Rule

±1σ = 68%

±2σ = 95%

±3σ = 99.7%

Example – Empirical Rule

•Scores on a standardized test are normalized with a mean of 500

•Assume a normal distribution with a standard deviation of 100

•What is the probability a randomly selected student’s score will be:▫More than 600▫Between 300 and 500▫Less than 400▫Between 400 and 700

Standard Normal Distribution

Standardizing Individual Data Values

• The standardized z-score is how far above or below the individual value is compared to the population mean in units of standard deviation.▫“How far above or below”= data value – mean▫“In units of standard deviation”= divide by s

© 2008 Thomson South-Western

ExampleThe average hotel check-in time is 12 minutes. Mary just left the cab that brought her to her hotel. Assuming a normal distribution with a standard deviation of 2.0 minutes, what is the probability that the time required for Mary and her bags to get to the room will be:

a) greater than 14 minutes?b) less than 8.5 minutes?c) between 10.5 and 14.0 minutes?

Example - CDF

•An average light bulb manufactured by the Acme Corporation lasts 300 days with a standard deviation of 50 days. Assuming that bulb life is normally distributed, what is the probability that an Acme light bulb will last at most 365 days?

•http://davidmlane.com/hyperstat/z_table.html

More Practice•The average charitable contribution

among people making $60,000 - $75,000 is $1935.

•Assume donations are normally distributed

•Assume a standard deviation of $400.▫What’s the probability that a randomly

selected person in this category made charitable contributions of at least $1600?

Normal Approximation of the Binomial•Continuity correction

▫Add or subtract .5 to correct for the gaps•Useable when:

▫nπ and n(1-π) are both >+5

Practice

•An expert claims there is no difference between the taste of 2 soft drinks.

•In a taste test involving 200 people, 55% of the testers preferred soft drink A.

• If the expert was correct, what’s the probability that 110 or more of the testers would prefer soft drink A?