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STA291

STA291. Statistical Methods Lecture 14. Last time …. We were considering the binomial distribution …. Continuous Probability Distributions. For continuous distributions, we cannot list all possible values with probabilities Instead, probabilities are assigned to intervals of numbers

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STA291

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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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