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Understanding Continuous Probability Distributions and the Normal Distribution

Learn about continuous probability distributions, the Normal Distribution, probability density functions, verification of empirical rules, normal probability calculations, and more in statistics.

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Understanding Continuous Probability Distributions and the Normal Distribution

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  1. STA291 Statistical Methods Lecture 14

  2. Last time … We were considering the binomial distribution …

  3. Continuous Probability Distributions • For continuous distributions, we cannot list all possible values with probabilities • Instead, probabilities are assigned to intervals of numbers • The probability of an individual number is 0 • Again, the probabilities have to be between 0 and 1 • The probability of the interval containing all possible values equals 1 • Mathematically, a continuous probability distribution corresponds to a probability densityfunction whose integral equals 1

  4. Continuous Probability Distributions: Example f(y) Area = 0.24 = P ( 7 < Y < 9 ) y 7 9 • Example: Y=Weekly use of gasoline by adults in North America (in gallons) • P(7 < Y < 9)=0.24 • The probability that a randomly chosen adult in North America uses between 7 and 9 gallons of gas per week is 0.24 • 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.

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

  6. Some Continuous Distributions

  7. The Normal Distribution • Carl Friedrich Gauß (1777-1855), Gaussian Distribution • Normal distribution is perfectly symmetric and bell-shaped • Characterized by two parameters: mean m and standard deviation s • 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. • The IQR  4/3s rule also applies

  8. Verifying Empirical Rule • The 68%-95%-99.7% rule can be verified using Z table in your (e) text • How much probability is within one (two, three) standard deviation(s) of the mean? • Note that the table only answers directly: How much probability is below a z of ±1, ±2, or ±3, for instance.

  9. Normal Distribution Table • 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. • For z =1.43, the tabulated value is 0.9236 • 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 • That is, the probability below μ + 1.43sof any normal distribution equals 0.9236

  10. Normal Probability Calculation • Forwards • 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 • Backwards • 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

  11. Looking back • Continuous distributions • notion of probability density function • many different examples • Normal, or Gaussian distribution • Standard normal • Forward and backward calculations

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