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Lecture

# UASTAT141Ch16-17.pdf

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School
University of Alberta
Department
Statistics
Course
STAT141
Professor
Paul Cartledge
Semester
Winter

Description
Ch. 16 – Random Variables Def’n: A random variable is a numerical measurement of the outcome of a random phenomenon. A discrete random variableis a random variable that assumes separate values. Æ # of people who think stats is dry The probability distribution of a discrete random variable lists all possible values that the random variable can assume and their corresponding probabilities. Notation: X = random variable; x = particular value; P(X = x) denotes probability that X equals the value x. Ex16.1) Toss a coin 3 times. Let X be the number of heads. 8 possible values: HHH, HHT, HTH, THH, HTT, THT, TTH, TTT 16abe0 x P(X = x) 0 0.125 1 0.375 2 0.375 3 0.125 Two noticeable characteristics for discrete probability distribution: 1. 0 ≤ P(X = x) ≤ 1 for each value of x 2. ∑ P(X = x) = 1 Ex16.2) Find the probabilities of the following events: “no heads”: P(X = 0) = 0.125 “at least one head”: P(X ≥ 1) = P(X = 1) + P(X = 2) + P(X = 3) = 0.375 + 0.375 + 0.125 = 0.875 “less than 2 heads”: P(X < 2) = P(X = 0) + P(X = 1) = 0.125 + 0.375 = 0.500 The population mean µ of a discrete random variable is a measure of the center of its distribution. It can be seen as a long-run average under replication. More precisely, µ = x P(X = x ) ∑ i i Sometimes referred to as µ = E(X) = the expected value of X. Keep in mind that µ is not necessarily a “typical” value of X (it’s not the mode). Ex16.3) Using Table 16X0, µ = ∑ x i(X = x i = (0)(0.125) + (1)(0.375) + (2)(0.375) + (3)(0.125) = 1.5 Æ On average, the number of heads from 3 coin tosses is 1.5. Ex16.4) Toss an unfair coin 3 times (hypothetical). Let X be as in previous example. x P(X = x) µ = x P(X = x ) ∑ i i 0 0.10 = (0)(0.10) + (1)(0.05) + 1 0.05 (2)(0.20) + (3)(0.65) 2 0.20 = 2.4 3 0.65 nd As 2 example shows, interpretation of µ as a measure of center of a distribution is more useful when the distribution is roughly symmetric, less useful when the distribution is highly skewed. The population standard deviation σ of a discrete random variable is a measure of variability of its distribution. As before, the standard deviation is defined as the square root of the population variance σ , given by 2 2 2 2 σ = ∑ (xi− µ) P(X = x )i= ∑ xiP(X = x )i− µ Ex16.5) From Table 16X0, 2 2 222 2 2 2 1 1 3 3 4963 5 . 1 ] =) ( 3 ) ( 2 ) ( = σ− µ1 ∑ ( i i) [ 8 8 8 8 8484 Ch. 17 – Probability Models Counting Techniques A permutation of a set is an ordered sequence of the elements in the set. n! = n × (n – 1) × (n – 2) × … × 2 × 1 Ex17.1) How many ways can the word COMEDY be arranged? 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 If r ordered elements of a set of size n are desired, then n n! P r (n− r)! Ex17.2) How many ways can 3 of the letters in COMEDY be arranged? P = 6! = 720 =120 3 (6 3)! 6 If order is NOT important when choosing r elements from a set of size n, then ⎛ ⎞ n! C r ⎜ ⎟ = ⎝ ⎠ r n r )! Ex17.3) How many ways can 3 letters be chosen from the word COMEDY? ⎛6 ⎞ 6! 720 ⎜ ⎟= = = 20 ⎝3 ⎠ 3!(6 3)! (6)(6) The Binomial Distribution Def’n: A Bernoulli trials a trial with only two possible outcomes, usually termed as a “success” and a “failure”. The prob. of success is p and the prob. of failure is 1 – p. Consider a random experiment consisting of n Bernoulli trials such that 1. The trials are independent. 2. Each trial results in only 2 possible outcomes: “success” and “failure”. 3. The probability of success in each trial (p) remains constant. Let an r.v. X be the number of successes in n trials where 0 < p < 1 and n = 1, 2, … Then, an X with these parameters has a binomial distribution such that ⎛⎞ x nx P())1== ( ⎜x − p x = 0, 1, …, n ⎝⎠ Also, µ = E(X) = np and σ =V(X))= np(1− p Ex17.4) Recall Oilers ticket phone line example from Ch. 15. Let X be the number of calls succeeding in making a sale. Thus, X ~ B(n, p) = B(10, 0.3). a) What is the probability that exactly 5 calls of 10 result in a sale? ⎛0 ⎞ 5 10−5 103 (X = 5) = ⎜ ⎟0.3 (1 0.3) = 0. ⎝5 ⎠ b) What is the probability that at least 3 calls result in a sale being made? P(X )≥1−( ) b) or P(X ≤ a). Ex17.5) Suppose we have a “uniform” distribution where obtaining each value
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