STA291 Statistical 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 0 o Again, the probabilities have to be between 0 and 1 o The probability of the interval containing all possible values equals 1 o 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)=0.24 o 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)f(y) y 7 9 Area = 0.24 = P ( 7 < Y < 9 )

Graphs for Probability Distributions o Discrete Variables: o Histogram o Height of the bar represents the probability o Continuous Variables: o Smooth, continuous curve o Area under the curve for an interval represents the probability of that interval

Some Continuous Distributions

The Normal Distribution o Carl Friedrich Gauß ( ), Gaussian Distribution o Normal distribution is perfectly symmetric and bell- shaped o Characterized by two parameters: mean  and standard deviation  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, o The IQR  4/3  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 o Distribution fact: this is the same as the probability that any normal with mean μ and standard deviation  takes a value below μ + z  o That is, the probability below μ  of any normal distribution equals

Normal Probability Calculation o Forwards o Given 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 Backwards o Given 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 o Continuous distributions o notion of probability density function o many different examples o Normal, or Gaussian distribution o Standard normal o Forward and backward calculations