Question

Show your work for each problem. 1) The probability of a driver will have an accident in 1 month equals .02. Find the probability that in 100 months he will have at least one accident. Find the solution exactly and approximately using another distribution. 2) Let X(t) Acoswt 120 where w is a constant and A is a uniform random variable over(0,1). The autocorrelation R, (5,1) is a) b) Coswt - s) cos(ws) cos(wt) 3 دیا 3) Suppose the probability of having a disease is .001, and a blood test is 100% effective in detecting the disease when the person has the disease. However, the test yields a "false positive" for 1% of healthy persons tested. What is the probability a person has the disease given that his test result is positive? Is the answer approximately a).44 b).01 c).09 d).90 4) Let G(x) be the cumulative distribution for a uniform random variable over (0,1) 0x<0 G(x)=x0sxsl. If the joint distribution of X and Y is F(x, y) = G(x)G(y), 1 x> a) find the joint density f(x, y) and b) P(x >Y). 5) Suppose independent random variables X have a uniform distribution on [i, i +1], i = 1,..24. Set X - į xEstimate P(310s X s 314) with the best method possible. 6) In the baseball World Series the first team to win 4 of 7 games wins the series and it stops. Suppose the Yankees play the Dodgers and the probability that either team will win is 0.5. What is the probability that the series will last for 7 games?

Answer 1

We would be looking at the first question here as:

Q1) We are given here that: P(accident) = 0.02

The probability that in 100 months, he will have at least one accident is computed here as:

= 1 - Probability of no accidents in 100 months

= 1 - (1 - 0.02)¹⁰⁰

= 1 - 0.98¹⁰⁰

= 0.8674

Therefore 0.8674 is the required probability here.