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Probability Distributions and Random Variables: Computation, Mean, and Variance

Learn about probability distributions, random variables, expected values, variance, mean, and more. Explore examples and computations for discrete and continuous distributions, including Bernoulli and linear functions.

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Probability Distributions and Random Variables: Computation, Mean, and Variance

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  1. Set 4 Probability distribution, expected value, variance

  2. Numerical Random Variables • Unknown quantities • Numerical values associated with outcomes • S = {TTT, TTH, THT, HTT, HHT, HTH, THH, HHH} X = Number of H’s in the outcome • S = { All students in a university} • X = GPA, 0 < x < 4 • Y = Gender, y=0 if male, y=1, if female

  3. Discrete probability distribution • Discrete random variables Y • Countable outcomes • Finite number of outcomes: -1, 0, 2 • Infinite number of outcomes: 0, 1, 2, ... • Also non-integers outcomes: -1.50, 0, 1, 1.5, 2, 2.75 • A probability mass functionf(y)=P(Y=y)directly gives probability of the outcome, y • f(y)tabular, needle (bar) graph, formula • Two properties of f(y): 0 < f(y) < 1 Sf(y) = 1

  4. An example

  5. Continuous probability distribution • Outcomes of random variableYare all numbers in an interval, Y = Income, y > 0; Event A={x: a < y < b } • The points in an interval are not countable • Probability of an outcome in an interval is given by the area under a function f(y) called probability density function P(a < Y < b)

  6. Density function f(y) • Yis continuous • Value of y can be any number in an interval • The points in an interval are not countable • f(y) is the equation of an idealized histogram • Density is always above or on the x-axis, f(y) > 0 • Total area under f(y) = 1 • The probability of the values of y in each interval a<y<b, is given by the area under f(y) • Proportion for any single value of y is zero

  7. Expected value of a random variable • Mean = Expected value of the variable • Discrete distribution E(Y) =Sy f(y) = m • Continuous distribution

  8. Interpretation of the mean • Center of the gravity of the distribution .5 .3 .2 y -1 0 my=.8 2 1.2 -1.8 -.8 0 • Deviations from the mean: y-m • Mean of deviations from the mean is is always zero

  9. Variance of a random variable • Variance = Expected value of the square deviations from the mean Var(Y) = E[(y - m )2 ]=s2 • Discrete distribution Var(Y) =S(y - m )2f(y) = s2 • Continuous distribution • Standard Deviation SD(y) =s

  10. ExampleComputation of mean and variance • Discrete case

  11. Bernoulli Model • Two outcomes • Success (Y=1)withP(Y=1)=p • Failure(Y=0)with P(Y=0)=1-p • Bernoulli distribution • Formulas for mean and variance

  12. Example • 10% of items produced by a machine are defective. • Y= 1, if an item is defective • Y= 0, otherwise • Bernoulli parameter p=.1.

  13. Median,Mean, Mode • Normal density function • Symmetric m Mean = Median = Mode

  14. Median,Mean, Mode • Skewed density function • Right skewed 50% 50% Mode Mean Median

  15. Interpretations • Mean • Weighted average, weight = probability • The balancing point (center of the gravity) • For symmetric distributions, mean = median • Near the long-run average of outcomes of f(y) • Law of large number • An expected value may be an impossible event , so we cannot expect to occur • Variance and Standard deviation • Measures spread of the distribution • Mean and variance are not informative for skewed distributions

  16. Percentiles • Cumulative distribution function CDF(a) = P(Y<a) • CDF(a) = .15, a is the 15th percentile • CDF(a) = .25, a is the 25th percentile (1st quartile, Q1) • CDF(a) = .50, a is the 50th percentile (median) • CDF(a) = .75, a is the 75th percentile (3rd quartile, Q3) • CDF(a) = .95, a is the 95th percentile • Interquartile Range: Q3- Q1

  17. Graph of a discrete CDF • Step function 1 .5 .2 -1 0 2

  18. Discrete CDF • Percentiles may not be unique • 20th percentile is any number in the interval -1<a <0 • Median is any number in the interval 0 <a < 2

  19. Continuous CDF • f(y) is density function • Percentiles are unique • Example: [Normal table gives CDF] • CDF(1.28) = P(Z < 1.28) = .90 .90 Density f(z) .5 CDF F(z) . 90 Height = Fz(1.28) Area =Fz(1.28) 1.28 1.28

  20. Linear function of a random variable • Given a quantitative random variable X • f(x), mx, sx • Another random variable Y is a linear function of X y = a + b x • Mean my = a + b mx • Also true for median, quartiles, percentiles • Variance of the linear function s2y = b2s2x • Standard deviation of the linear function sy = |b|sx • Also true for inter-quartile range

  21. Examples • Given a random variable X with mean and variance • Compute mean, variance and standard deviation of Y = 5 + 2X • Compute mean, variance and standard deviation of V = 5 – 2X

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