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Math 1180 Summary of topics and practice problems for Midterm 3 1. Relationshp between two random variables - Compute joint distributions from marginal and conditional distributions (7.1) - Compute mrginal and conditional distributions from a joint distrbution (7.1) - check joint distribution for independence - Compute covariance (7.2) - Compute correlation (7.2) - Interpret near-zero, near-one, near-negative-one correlation (7.2) 2. Expected value, variance and covariance of random variable combinations (7.3) - Expected value of a sum - expected value of a product - varince of a sum - expected value, variance and covariance of variables multiplies by a constant 3. Binomial distribution (7.5) 4. Geometric distribution (7.6) 5. POisson distribution (7.7) 6. Exponential distribution (7.6) For each distribution you need to be able to: - Recognize the distribution in a variety of settings - Computing the correpsonding probabilities using the formulas - Finding the mean and variance 7. Normal distribution (7.8,7.9) - Computing probabilities from Normal distribution by reducing to the Standard Normal probabilities and using the table - Marking probabilities on the graph - Find distributions for sums and averages of random variables using Central Limit Theorem 8. Approximations - Use POisson distribution to compute binomial probabilities - Use normal distribution to compute poisson and binomial probabilities 1. A staining technique has a 20% chance of success. Let Sn be the ranom variable representing the number of successful staining in n attempts. a) What are the conditions for Sn to have a binomial distribution? b) Under the binomial distribution conditions, find expected value and standard deviation of S10 c) Find P r(S10 = 2), using the fact that 10 = 45 2 2. Each of two guitarists only knows A and D chords. On each beat each guitarist plays an A chord with probability 0.75, and a D chord with probablity 0.25. These two probabilities are assumed the same in all subsequent questions. a) If they begin by playing independently, give the joint distribution b)If the second player plays an A with probability 90% when the second player does, give the joint distribution c) For the following joint distribution, find marginal distributions. P r(AA) = 0.1, P r(AD) = 0.5, P r(DA) = 0.3, P r(DD) = 0.1 d) If playing A is worth 1 point and playing D is worth 0, find the covariance, and correlation in case of distribution from c) 3. A study finds that the ice cream sales and the rate of drowning deaths have a correlation of .85. a) Is this a strong correlation? b) Based on this information what would you predict happened with ice cream sales last month if we know that the rate of drowning deaths increased sharply? c) Can we conclude that ice cream consumption causes drowning? 4. N1 is the number of successful shooting attempts by player F. He usually hits the target with probability 0.8. N2 is the number of successful attempts by player G who hits each target with probability 0.4. a) If player F has 10 attempts, and player G has 3 attempts, what is TOTAL expected number of successful hits from both players? b) If player F gets 3 points for each of the hits, what is the expected number and variance of number of points he will earn? c) If the player F still gets 3 points, and the player G gets 2 points for each time he scores, what is the expected number of TOTAl points they will get together? If the players shoot independently, what is the variance of the TOTAl score? If the independency assumption is removed, can you compute the variance? 5. On average, there are 222 sunny days per year in Salt Lake City. Assuming that the weather on each day is independent of other days, and that any day can be sunny with the same probability, find probability that at least 2 days among the next 10 will be sunny. Today is rainy. How long will I have to wait on average to the next rainy day? 6. If a student scores 10 on the exam with probability .1, scores 5 with probability .7 and scores 0 with probability .2, find the probability that an average grade in 100 student class is above 5, using central limit theorem. 7. Action potentials of a neuron can be thought of as a POisson process with rate of 20 Hz. Find probability that the time between second and third action potential is longer than 0.01 but shorter than 0.06 seconds. How many spikes to we expect in a 0.5 sec interval? What is the probability that the number of spikes in a 0.5 sec interval is less than expected? 8. A measurement result is normally distributed with mean 0.1 and standard eviation 0.05. How often will we get a negative measurement? (use the table at the back of the book) What is the probability that the measurement falls between 0.15 and 0.2? Mark associated probability on the graph of the pdf. Formulae list P r(A|B) = P r(A ∩ B) P r(B) P r(A ∩ B) = P r(A|B)P r(B) Pr(A) = Σni=1 P r(A|Ei)P r(Ei) P r(A|B)P r(B) P r(A) P r(B|A) = X̄ = E(X) = Σni=1 xi p(xi ) X̄ = E(X) = Z b a xf (x)dx MAD = Σni=1 |xi − X̄|p(xi ) MAD = Z b |x − X̄|f (x)dx a V ar = σ 2 = Σni=1 x2i p(xi ) − X̄ 2 V ar = σ 2 = Cov(X, Y ) = Z b a m X n X j=1 i=1 x2 f (x)dx − X̄ 2 (xi − X̄)(yj − Ȳ )pij Cov(X, Y ) = E(XY ) − X̄ Ȳ ρX,Y = Cov(X, Y ) σX σY E(X + Y ) = E(X) + E(Y ) E(aX) = aE(X) E(XY ) = E(X)E(Y ) + Cov(X, Y ) V ar(X + Y ) = V ar(x) + V ar(Y ) + 2Cov(X, Y ) V ar(aX) = a2 V ar(X) Cov(aX, Y ) = Cov(x, aY ) = a · Cov(X, Y ) ! n k p (1 − p)n−k P r(N = k) = b(k; n, p) = k E(N) = np, V ar(N) = np(1 − p) P r(T = t) = gt = p(1 − p)t−1 P r(T ≤ t) = Gt = 1 − (1 − p)t 1 1−p E(T ) = , V ar(t) = p p2 (λt)k e−λt k! E(N) = λt, V ar(N) = λt P r(N = k) = p(k; λ, t) = T : f (t) = λe−λt ; F (t) = 1 − e−λt E(T ) = 1 1 , V ar(T ) = 2 λ λ Sn ∼ N(nµ, nσ 2 ) An ∼ N(µ, σ 2 ) X −µ σ (x−µ)2 1 e− 2σ2 f (x) = √ 2πσ 2 Z= A copy of table from the book