1. An insurance company sells several types of insurance policies, including auto policies and homeowner policies....

Transcribed Image Text:

1. An insurance company sells several types of insurance policies, including auto policies and homeowner policies. Let A₁ be those people with an auto policy only, A₂ those people with a homeowner policy only, and A3 those people with both an auto and homeowner policy (but no other policies). For a person randomly selected from the company's policy holders, suppose that P(A₁) = 0.3, P(A₂) = 0.2, and P(A3) = 0.2. Further, let B be the event that the person will renew at least one of these policies. Say from the past experience that we assign the conditional probabilities P(B|A₁) = 0.6, P(B|A₂) = 0.7, and P(B|A3) = 0.8. Given that the person selected at random has an auto or homeowner policy (or both), what is the conditional probability that the person will renew at least one of those policies? 2. A, B, and C are events with P(A) = 0.3, P(B) = 0.4, P(C) = 0.5, A and B are disjoint, A and C are independent, and P(B|C) = 0.1. Find P(A U BU C). 3. If P(C₁) > 0 and if C2, C3, C4, ... are mutually disjoint sets, show that P(C2 U C3 U...|C1) = P(C2|C1) + P(C3|C1) + ... 4. Let A and B be two events. a) If the events A and B are mutually exclusive, are A and B always independent? If the answer is no, can they ever be independent? Explain. b) If A c B, can A and B ever be independent events? Explain. 5. A Pap smear is a screening procedure used to detect cervical cancer. For women with this cancer, there are about 16% false negatives; that is P(T- test negative | C+ = cancer) = 0.16. For women without cancer, there are about 10% false positives; that is P(T+ = test positive | C = not cancer) = 0.10. There are about 8 women in 100,000 who have this cancer. What is the probability of having the cervical cancer given a positive test? 6. Let X be a random variable with probability mass function (pmf) fx(x) = c(2/3)x if x = 1, 2, 3,..., and = 0 elsewhere. a) Find the constant c so that fx(x) satisfies the condition of being a pmf of the random variable X. b) Find P(X= 1 or 2) c) Find P(1/2 < X < 5/2). d) Find P(1 ≤ x ≤ 2). e) Find the cumulative distribution function (cdf) Fx(x) of X. 7. Given the cdf Fx(x) 0 if x < -1 X+2 if -1 < x < 1 4 1 if x ≥ 1 of a random variable X, compute a) P(-1/2 < X ≤ 1/2) b) P(X=0) c) P(X= 1) d) P(2 < x≤ 3) 8. Let X have the pmf fx(x) = 1/3 if x = -1, 0, 1. Find the pmf of Y = X². 9. Let X be a continuous random variable with pdf fx(x) = 2xexp(-x²), a) Find the pdf of Y = X². b) Compute the probability that Y is at least 2 given that Y is at least 1. 10. The probability mass function (pmf) of a random variable X is given by fx(x) = x = 1,2,3,... a) Find the moment generating function of X. b) Find the mean and variance of X. c) Find the pmf of Y = X². 0<x<x 11. The probability density function (pdf) of a continuous random variable X is given by fx(x) = cx², 0<x< 1. a) Show that the value of c is 3. b) Find P(0.5 < X < 2). c) Find E(X³). d) Find Var(5x+10). e) Find the pdf of Y = X³. 12. The probability density function of a continuous random variable X is given by fx(x)=xe *, x>0 a) Find the moment generating function of X. b) Find the moments of the distribution of X. 1. An insurance company sells several types of insurance policies, including auto policies and homeowner policies. Let A₁ be those people with an auto policy only, A₂ those people with a homeowner policy only, and A3 those people with both an auto and homeowner policy (but no other policies). For a person randomly selected from the company's policy holders, suppose that P(A₁) = 0.3, P(A₂) = 0.2, and P(A3) = 0.2. Further, let B be the event that the person will renew at least one of these policies. Say from the past experience that we assign the conditional probabilities P(B|A₁) = 0.6, P(B|A₂) = 0.7, and P(B|A3) = 0.8. Given that the person selected at random has an auto or homeowner policy (or both), what is the conditional probability that the person will renew at least one of those policies? 2. A, B, and C are events with P(A) = 0.3, P(B) = 0.4, P(C) = 0.5, A and B are disjoint, A and C are independent, and P(B|C) = 0.1. Find P(A U BU C). 3. If P(C₁) > 0 and if C2, C3, C4, ... are mutually disjoint sets, show that P(C2 U C3 U...|C1) = P(C2|C1) + P(C3|C1) + ... 4. Let A and B be two events. a) If the events A and B are mutually exclusive, are A and B always independent? If the answer is no, can they ever be independent? Explain. b) If A c B, can A and B ever be independent events? Explain. 5. A Pap smear is a screening procedure used to detect cervical cancer. For women with this cancer, there are about 16% false negatives; that is P(T- test negative | C+ = cancer) = 0.16. For women without cancer, there are about 10% false positives; that is P(T+ = test positive | C = not cancer) = 0.10. There are about 8 women in 100,000 who have this cancer. What is the probability of having the cervical cancer given a positive test? 6. Let X be a random variable with probability mass function (pmf) fx(x) = c(2/3)x if x = 1, 2, 3,..., and = 0 elsewhere. a) Find the constant c so that fx(x) satisfies the condition of being a pmf of the random variable X. b) Find P(X= 1 or 2) c) Find P(1/2 < X < 5/2). d) Find P(1 ≤ x ≤ 2). e) Find the cumulative distribution function (cdf) Fx(x) of X. 7. Given the cdf Fx(x) 0 if x < -1 X+2 if -1 < x < 1 4 1 if x ≥ 1 of a random variable X, compute a) P(-1/2 < X ≤ 1/2) b) P(X=0) c) P(X= 1) d) P(2 < x≤ 3) 8. Let X have the pmf fx(x) = 1/3 if x = -1, 0, 1. Find the pmf of Y = X². 9. Let X be a continuous random variable with pdf fx(x) = 2xexp(-x²), a) Find the pdf of Y = X². b) Compute the probability that Y is at least 2 given that Y is at least 1. 10. The probability mass function (pmf) of a random variable X is given by fx(x) = x = 1,2,3,... a) Find the moment generating function of X. b) Find the mean and variance of X. c) Find the pmf of Y = X². 0<x<x 11. The probability density function (pdf) of a continuous random variable X is given by fx(x) = cx², 0<x< 1. a) Show that the value of c is 3. b) Find P(0.5 < X < 2). c) Find E(X³). d) Find Var(5x+10). e) Find the pdf of Y = X³. 12. The probability density function of a continuous random variable X is given by fx(x)=xe *, x>0 a) Find the moment generating function of X. b) Find the moments of the distribution of X.

Answer rating: 100% (QA)

Ang Given data PA 03 PA 02 P A3 02 PB1A 06 P BA 0... View the full answer