STAT101_Chap4

# STAT101_Chap4 - 4 Probability Distributions Probability...

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4. Probability Distributions Probability : With random sampling or a randomized experiment, the probability an observation takes a particular value is the proportion of times that outcome would occur in a long sequence of observations. Usually corresponds to a population proportion (and thus falls between 0 and 1) for some real or conceptual population.

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Basic probability rules Let A, B denotes possible outcomes P(not A) = 1 – P(A) For distinct possible outcomes A and B, P(A or B) = P(A) + P(B) P(A and B) = P(A)P(B given A) For “independent” outcomes, P(B given A) = P(B), so P(A and B) = P(A)P(B).
Happiness (2008 GSS data) Income Very Pretty Not too Total ------------------------------- Above Aver. 164 233 26 423 Average 293 473 117 883 Below Aver. 132 383 172 687 ------------------------------ Total 589 1089 315 1993 Let A = average income, B = very happy P(A) estimated by (a “marginal probability”), P(not A) = 1 – P(A) = P(B given A) estimated by (a “conditional probability”) P(A and B) = P(A)P(B given A) est. by (which equals , a “joint probability”)

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B1: randomly selected person is very happy B2: second randomly selected person is very happy P(B1), P(B2) estimated by P(B1 and B2) = P(B1)P(B2) estimated by If instead B2 refers to partner of person for B1, B1 and B2 probably not independent and this formula is inappropriate
Probability distribution of a variable Lists the possible outcomes for the “random variable” and their probabilities Discrete variable : Assign probabilities P(y) to individual values y, with 0 ( ) 1, ( ) 1 P y P y Σ =

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Example: Randomly sample 3 people and ask whether they favor (F) or oppose (O) legalization of same-sex marriage y = number who “favor” (0, 1, 2, or 3) For possible samples of size n = 3, Sample y Sample y (O, O, O) 0 (O, F, F) 2 (O, O, F) 1 (F, O, F) 2 (O, F, O) 1 (F, F, O) 2 (F, O, O) 1 (F, F, F) 3
If population equally split between F and O, these eight samples are equally likely and probability distribution of y is y P(y) 0 1 2 3 (special case of “binomial distribution,” introduced in Chap. 6). In practice, probability distributions are often estimated from sample data, and then have the form of frequency distributions

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Example : GSS results on y = number of people you knew personally who committed suicide in past 12 months (variable “suiknew”). Estimated probability distribution is y P(y) 0 .895 1 .084 2 .015 3 .006
Like frequency distributions, probability distributions have descriptive measures, such as mean and standard deviation Mean ( expected value ) - ( ) ( ) E Y yP y μ= = µ = represents a “long run average outcome” (median = mode = 0)

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Standard Deviation - Measure of the “typical” distance of an outcome from the mean, denoted by σ (We won’t need to calculate this formula.) If a distribution is approximately bell-shaped, then: all or nearly all the distribution falls between µ - 3σ and µ + 3σ Probability about 0.68 falls between µ - σ and µ + σ 2 = ( ) ( ) y P y σ μ Σ -
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STAT101_Chap4 - 4 Probability Distributions Probability...

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