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Discrete Random Variables Discrete list of distinct values. Typically used when a variable is integer-valued without too many possible choices. choices pdf (Probability distribution function) For example: P(X = 2) = 0.278 If X is the sum of two fair dice when rolled. Continuous Random Variable Assumes a range of values covering an interval. _____________. May be limited by instrument’s accuracy / decimal points, but still continuous. is this area Find probabilities using a probability density function, which is a curve. Calculate probabilities by finding the area under the curve. • We can’t find probabilities for exact outcomes. • For example: P(X = 2) = 0. • Instead we can find probabilities for a range of values. 3 Expected Value = Sum of “value × probability” over all possible values Faculty Example: Calculate the Expected Value E(X) k P(X = k) 0 0.1 1 0.3 2 0.4 3 0.2 Total 1.0 X = # courses/semester taught by PSU faculty E(X) = µ = 0×(.1) + 1×(.3) + 2×(.4) + 3×(.2) = 1.7 Interpretation: • The average is 1.7 classes for this population Conditions for a binomial experiment 1 There are n “trials”, where n is fixed and known in advance 2 We can define two possible outcomes for each trial: “Success” (S) and “Failure” (F) 3 The outcomes are independent; no single outcome influences any other outcome 4 The probability of “Success” is the same for each trial. We use “p” to write P(Success). Mean and standard deviation for binomial random variables Mean: Standard deviation: How to relate all this to Z-scores • We can standardize values from any normal distribution to relation them to the standard normal distribution. Value Mean z Standard Deviation