STA291Statistical Methods

Lecture 14

Last time …

We were considering the binomial distribution …

2

Continuous Probability Distributions

o For continuous distributions, we cannot list all possible values with probabilities

o Instead, probabilities are assigned to intervals of numbers

o The probability of an individual number is 0o Again, the probabilities have to be between

0 and 1o The probability of the interval containing all

possible values equals 1o Mathematically, a continuous probability

distribution corresponds to a probability density function whose integral equals 1

Continuous Probability Distributions: Example

o Example: Y=Weekly use of gasoline by adults in North America (in gallons)

o P(7 < Y < 9)=0.24o The probability that a randomly chosen adult in

North America uses between 7 and 9 gallons of gas per week is 0.24

o Probability of finding someone who uses exactly 7 gallons of gas per week is 0 (zero)—might be very close to 7, but it won’t be exactly 7.

f(y)

y7 9

Area = 0.24 = P ( 7 < Y < 9 )

Graphs for Probability Distributions

o Discrete Variables:oHistogramoHeight of the bar represents the

probability

o Continuous Variables:oSmooth, continuous curveoArea under the curve for an interval

represents the probability of that interval

Some Continuous Distributions

The Normal Distribution

o Carl Friedrich Gauß (1777-1855), Gaussian Distribution

o Normal distribution is perfectly symmetric and bell-shaped

o Characterized by two parameters: mean m and standard deviation s

o The 68%-95%-99.7% rule applies to the normal distribution; that is, the probability concentrated within 1 standard deviation of the mean is always (approximately) 0.68; within 2, 0.95; within 3, 0.997.

o The IQR 4/3s rule also applies

Verifying Empirical Rule

o The 68%-95%-99.7% rule can be verified using Z table in your (e) text

o How much probability is within one (two, three) standard deviation(s) of the mean?

o Note that the table only answers directly: How much probability is below a z of ±1, ±2, or ±3, for instance.

Normal Distribution Table

o Table Z shows for different values of z the probability below a value of z, a normal that has mean 0 and standard deviation 1.

o For z =1.43, the tabulated value is 0.9236

o Distribution fact: this is the same as the probability that any normal with mean μ and standard deviation s takes a value below μ + zs

o That is, the probability below μ + 1.43s of any normal distribution equals 0.9236

Normal Probability Calculationo ForwardsoGiven a value or values from the

distribution, we wish to find the proportion of the entire population that would fall above/below/ between the values

o BackwardsoGiven a “target”

probability/proportion/ percentage, we wish to find the value or values from the distribution that delimit that fraction of the population, either above, below, or between them

Looking back

oContinuous distributionso notion of probability

density functiono many different examplesoNormal, or Gaussian distributionoStandard normaloForward and backward calculations