Do HW in Probability and Statistics

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Do HW in Probability and Statistics

MATH 331 sec002 – Fall 2016 – Homework 5 Instructor: Dr. Brown 1. Precisely ten thousand people will play a certain lottery each day. Each lottery ticket costs $1. Every day, 100 people are chosen as ?winners?. Of those 100 people, 80 will win $2, 19 will win $100, and one lucky winner will win $7,000! Let X denote the profit (= winnings - cost) of a randomly chosen player of this game. (a) Give both the range and probability mass function for X. (b) Find Pr(X ≥ 100). (c) Compute E(X) and Var(X) 2. Continuing Problem 1, Let Z be the profit of the casino that is providing the lottery. (a) Give both the range and probability mass function for Z. (b) Compute E(Z) and Var(Z). 3. In each of the following cases find all values of b for which the given function is a probability density function. a. f (x) =  2  x − b,  b.  g(x) = cos(x) 0, 0, if 1 ≤ x ≤ 3, otherwise. if − b ≤ x ≤ b, otherwise. 4. Suppose that the random variable X has cumulative distribution function given by  0, if x < 1,        1  , if 1 ≤ x < 32 ,    3 F (x) = 1 , if 32 ≤ x < 53 ,  2       3  , if 53 ≤ x < 95 ,    4 1, if x ≥ 59 , a. Find the range of X. b. Show that X is discrete, and find its probability mass function. 5. Choose a point uniformly at random in a unit square (square of side length 1). Let X be the distance from the point chosen to the nearest edge of the square. a. Find the cumulative distribution function of X. b. Find the probability density function of X. 6. Show that if the random variable X only takes nonnegative integers as its values then E(X) = ∞ X Pr(X ≥ k). k=1 This holds even when E(X)P = ∞, in which case the sum on the right hand side is infinite. Hint: Write Pr(X ≥ k) as ∞ i=k Pr(X = i) in the sum, and then switch the order of the two summations. 7. a. Let X ∼ Geom(p). Compute Pr(X ≥ k) for all positive integers k. b. Use Exercise 6. and part (a) to give a new proof that the expected value of a Geom(p) random variable is p−1 . 1+q . p2 Hint: To compute this conveniently, start with E(X 2 ) = E(X) + E[X(X − 1)]. We learned that the mean of a sum is always the sum of the means, but for now we can easily justify this claim by breaking up the sum: P P E(X 2 ) = k k 2 Pr(X = k) = k (k + k(k − 1)) Pr(X = k) 8. Let X ∼ Geom(p) and q = 1 − p. Derive the formula E(X 2 ) = = P k k Pr(X = k) + P k k(k − 1) Pr(X = k) = E(X) + E[X(X − 1)]. Then note that in the geometric case E[X(X − 1)] can be computed with a derivative 00 technique ((xk ) = k(k − 1)xk−2 for k ≥ 2) 9. Suppose a random variable X has density  −4 cx , f (x) = 0, if x ≥ 1, otherwise. where c is a constant. a. What must be the value of c? b. Find Pr(0.5 < X < 1). c. Find Pr(0.5 < X < 2). d. Find Pr(2 < X < 4). e. Find the cumulative distribution function FX (x). f. Find E(X) and Var(X). 10. Suppose that the random variable X has expected value E(X) = 3 and variance Var(X) = 4. Compute the following quantities. Recall that E(aX + b) = aE(X) + b, Var(aX + b) = a2 Var(X), Var(X) = E[X 2 ] − (E[X])2 . a. E[3X + 2]. b. E[X 2 ]. c. E[(2X + 3)2 ]. d. Var[4X − 2].
MATH 331 sec002 – Fall 2016 – Homework 6 Instructor: Dr. Brown 1. Let X be a normal random variable with mean µ = −2 and variance σ 2 = 7. Find the following probabilities. (a) Pr(X > 3.5). (b) Pr(−2.1 < X < −1.9). (c) Pr(X < 2). (d) Pr(X < −10). (e) Pr(X > 4). 2. Let X be a normal random variable with mean 3 and variance 4. (a) Find the probability Pr(2 < X < 6). (b) Find the value c such that Pr(X > c) = 0.33. (c) Find E(X 2 ) Hint: You can integrate with the density function, but it is quicker to relate E(X 2 ) to the mean and variance. 3. My bus is scheduled to depart at noon. However, in reality the departure time varies randomly, with average departure time 12 o’clock noon and a standard deviation of 6 minutes. Assume the departure time is normally distributed. If I get to the bus stop 5 minutes past noon, what is the probability that the bus has not yet departed? 4. In a lumberjack competition, a contestant is blindfolded and then spun around 9 times. The contestant then tries to hit the target point in the middle of a horizontal log with an axe (while still blindfolded). The contestant receives 15 points if his hit is within 3 cm of the target, 10 points if his hit is between 3 cm and 10 cm off the target, 5 points if his hit is between 10 cm and 20 cm off the target, and zero points if his hit is beyond 20 cm away from the target (and someone may lose a finger!). Let Y record the position of the hit, so that Y = y > 0 corresponds to missing the target point to the right by y cm and Y = −y < 0 corresponds to missing the target to the left by y cm. Assume that Y is normally distributed with mean µ = 0 and variance 100cm2 . Find the expected number of points that the contestant wins. 5. Assume 20% of a population prefers cereal A to cereal B. I go and interview 100 randomly chosen individuals. Use a normal approximation to estimate the probability that more than 25 people in my sample prefer cereal A to cereal B. (Whether you want to use the continuity correction is up to you.) 6. Suppose that the distribution of the lifetime of a car battery, produced by a certain car company, is well approximated by a normal distribution with a mean of 1.2 × 103 hours and variance 104 . What is the approximate probability that a batch of 100 car batteries will contain at least 20 whose lifetimes are less than 1, 100 hours? 7. A pollster would like to estimate the fraction p of people in a population who intend to vote for a particular candidate. How large must a random sample be in order to be at least 95% certain that the fraction p̂ of positive answers in the sample is within 0.02 of the true p? 8. A political interest group wants to determine what fraction p ∈ (0, 1) of the population intends to vote for candidate A in the next election. 1,000 randomly chosen individuals are polled. 457 of these indicate that they intend to vote for candidate A. Find the 95% confidence interval for the true fraction p. 9. In a million rolls of a biased die the number 6 shows up 180,000 times. Find a 99.9% confidence interval for the unknown probability that the die rolls six.

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