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E CON 310. TA S ECTION - W EEK 9 ´ prepared by S ANG Y OON (T IM ) L EE and I GNACIO M ONZ ON I LLN and CLT For a sequence of iid random variables { Xi }in=1 with mean µ and variance σ2 , the sum Sn and ¯ sample mean Xn are: n n 1 1 Sn = ∑ Xi , ¯ Xn = n ∑ Xi = n Sn i =1 i =1 and E(Sn ) = nµ, ¯ E ( Xn ) = µ Var (Sn ) = nσ2 , ¯ n ) = 1 σ2 Var ( X n The sample mean is crucial for LLN and CLT. Law of Large Numbers: P ( Xn ) ∈ (µ − , µ + )) → 1 as n → ∞. ¯ Coin ﬂips: ﬂip twice, don’t necessarily get exactly one heads and one tails. But when you ﬂip it more and more times, you do expect that you’ll get heads about one half of the time. Unfortu- nately, LLN does not tell you exactly how many times you have to do this for that to happen. LLN tells us that the sample mean converges to some number. On the other hand, CLT tells us that the sample mean converges to some distribution: Deﬁnition: For a sequence of random variables {Yn }∞=1 and a continuous random variable L, n the sequence converges to L in distribution, and we write “Yn →d L”, if for any interval ( a, b), P (Yn ∈ ( a, b)) → P ( L ∈ ( a, b)) as n → ∞ Coin ﬂips again: ﬂip the coin 10 million times, count the fraction of heads we get, do that again 1 and again 10 million times. Histogram will look much like a normal distribution centered at 2 : Central Limit Theorem: For a sequence of iid random variables { Xi }in=1 with mean E( Xi ) = µ and variance Var ( Xi ) = σ2 , Sn − nµ √ ¯ Xn − µ √ = n →d Z as n → ∞ nσ σ where Z ∼ N (0, 1). 2 ¯ For n large enough, Sn ≈ N (nµ, nσ2 ) and Xn ≈ N (µ, σ ). As with LLN, CLT does not tell us how n large n should be for this to be true. 1 VS Chapter 9, Exercise 6 An IRS computer ﬂags suspicious tax returns, which are then looked over by an IRS agent to determine whether an audit is needed. The probability that a ﬂagged return requires an audit is .40. Suppose that Agent Anderson evaluates 100 ﬂagged returns per week. a. What is the approximate probability distribution for S100 , the number of ﬂagged returns that require an audit this week? b. Determine the probability that this week, Agent Anderson ﬁnds between 35 and 40 returns (inclusive) that require an audit. c. Determine the probability that Agent Anderson ﬁnds more than 50 returns that require an audit. VS Chapter 9, Exercise 10 Two sales reps, Amy and Beatrice, work for a large shoe and apparel manufacturer. Their company is trying to convince retailers to allocate space to a new product line. The sales rep that secures the most new contracts with retailers this month earns a $10,000 bonus. Amy will convince a retailer to accept the new product line with a probability of .25, while Beatrice is successful with probability .30. Each rep’s sales calls can be modeled a sequence of iid random variables, and both reps have 100 retailers in their sales territory. a. Describe the approximate probability distributions of S A and SB , the total numbers of suc- cessful sales calls by Amy and Beatrice. b. Determine the probabilities that each rep has between 25 and 30 successful sales calls. c. What does the random variable S A − SB represent? What is its approximate distribution? d. What is the probability that Amy wins the bonus? 2 II Exponential Distribution Normal distribution: Bell curve, exponential distribution: waiting times. Time you wait for a bus, the next customer, someone else to drop the course so you can enroll... II.1 Deﬁnition The distribution function of a exponential random variable T is completely characterized by one parameter only: λ exp(−λt) if t ≥ 0 f (t) = 0 otherwise. λ describes how long you are likely to wait. λ large means shorter waiting time. II.2 Traits If T ∼ exp(λ), 1. P( T ≤ t) = 1 − exp(−λt), P( T > t) = exp(−λt) 1 1 2. E( T ) = λ , Var ( T ) = λ2 3. If Y = cT, then Y ∼ exp λ c . 4. Memorylessness: P( T > s + t| T > s) = P( T > t). Memorylessness might not come immediately. For example, you arrived at a bus stop and think a bus might come in about 2 minutes. After a minute, you still think the bus might come in about 2 minutes. 3 II.3 VS Chapter 8, Exercise 6 Your mail order ﬁrm employs a large number of operators to take phone orders. When Alvin begins a phone order the amount of time it takes for him to complete the order follows an ex- ponential distribution with rate .25, so that his expected time to complete an order is 4 minutes. Similarly, Bertha completes orders at an exponential rate of .2, and Cedric completes orders at an exponential rate of .18. The amounts of time it takes to complete orders are independent across operators. a. Suppose that Alvin begins a phone order, and that after 5 minutes the order is not yet com- plete. Conditional on this event, what is the probability that he will not complete the order until 10 or more minutes have passed in total? b. What is the probability that Alvin completes his next order in 2 minutes or less? What about Bertha? Cedric? Suppose all three operators begin taking an order simultaneously. c. What is the probability that the ﬁrst operator completes his or her order in 2 minutes or less? d. What is the probability that the ﬁrst ﬂashlight to die dies within six months? e. What is the probability that Bertha completes her order ﬁrst, doing so in 1.5 minutes or less? (Hint: Use the fact that the time required to complete the ﬁrst order and the identity of the operator who completes it are independent of one another.) II.4 VS Chapter 8, Exercise 7 John owns two ﬂashlights, one amber and one blue. The lifetime of each ﬂashlight has an expo- nential distribution. The expected lifetime of the amber ﬂashlight is 1.5 years, while the expected lifetime of the blue ﬂashlight is 2.5 years. a. What is the probability of this combination of events: the amber ﬂashlight lasts for less than 3 years, and the blue ﬂashlight lasts for more than 3 years. b. Suppose that the blue ﬂashlight lasts for 10 or more years. What is the probability that it lasts between 9 and 12 years in total? c. What is the probability that the amber ﬂashlight lasts longer than the blue ﬂashlight? d. What is the probability that the ﬁrst ﬂashlight to die dies within six months? 4