Let probability that a child will be born with black eyes is 1/3, determine the probability that only the fifth child of a family will be born with black eyes, and find its E(X), Var(X).
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- Suppose that during rainy season on a tropical island the length of theshower has an exponential distribution, with parameter λ = 2, time beingmeasured in minutes. What is the probability that a shower will last morethan three minutes? If a shower has already lasted for 2 minutes, what is theprobability that it will last for at least one more minute?Suppose we roll 2 six-sided dice. Let X be their sum and Y be the larger value of the two. (a) (10 points) What is E[X]? (b) (10 points) What is E[Y]?Assume that there is a rare disease. It occurs in the population at a rate of 1 per 1000; that is, P[disease] = 0.001. There is a test for the disease that has an error rate of 5% for in positive and negative senses. That is, if a patient has the disease, there is 95% probability that the test will confirm that status and 5% probability that the test will be negative. Conversely, if the patient does not have the disease, there is 5% probability that the test will indicate erroneously that the disease is present and 95% probability that the test will confirm that the disease is not present. Let PT indicate a positive test, D indicate the disease is present, and a bar over a symbol indicates the negative, so this can all be summarized as Po(D) = 0.001 P(PT|D) = 0.95 P(PT|D) = 0.95 P(PT|D) = 0.05 P(PT|D) = 0.05 The disease is serious, and the treatment is difficult and expensive, so a decision either to treat or not to treat the disease is not trivial. A patient takes the test and…
- Answer and provide right answer with complete solutionSuppose a basketball player who makes a free throw shot with success 80%. Let X be the number of the successful shots among n = 5 shots. (a) What is the probability that a basketball player makes exactly 4 shots successfully, or, P (X = 4)? (b) W h a t i s P (X = 5)? (c) Calculate P (X < 4), that is, the probability that this player makes less than 4 successful shots.gn X Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is 0.41 (a couple will travel together in the same vehicle, so either both people will be on time or else both will arrive late). Assume that different couples and individuals are on time or late independently of one another. Let X = the number of people who arrive late for the seminar. (a) Determine the probability mass function of X. [Hint: label the three couples #1, #2, and #3 and the two individuals #4 and #5.] (Round your answers to four decimal places.) P(X = X) X 0 1 2 3 4 5 6 7 8 (b) Obtain the cumulative distribution function of X. (Round your answers to four decimal places.) F(x) X 0 1 2 3 4 5 6 7 Use the cumulative distribution function of X to calculate P(2 ≤ x ≤ 5). (Round your answer to four decimal places.) P(2 ≤ x ≤ 5) = Need Help? Read It Watch It 4
- In 1938, a physicist named Frank Benford discovered that the number 1 appears in the first digit of random data more often than the number 2, the number 2 more often than the number 3 and so on. In general, the probability of occurrence of the first digit of a number can be written in the form of a probability function x + 1 P(X = x) = log. X a. Prove it P(X = x) = log ) untuk x = 1,2,3,4...,9 x+1 X x = 1,2,3,4..., 9 is a probability mass function 2 b. Find the cumulative distribution function of X!Suppose that you start with an initial fortune of 20 dollars, and you bet one dollar each time. The probability of winning each hand is p. You quit if you reach your goal of L dollars or when you go broke. Let Q(xo) denote that probability that you eventually win. We need to characterize this probability. Next we give a more mathematical description of the problem. Now define your fortune at time n ≥ 1 by - Xn = Xn-1+ I{Y=1} = I{Yn=0} with the initial condition X = x0 ≥0. Let L = N be given, and define T inf{k : Xk = 0 or Xk = L}. Finally, let Q(x0) = P{XT = . L}. Show that X is a Markov chain and obtain its transition probability matrix. Is it irreducible? Use conditional expectations to prove that Q(x)=pQ(x+1) + (1 − p)Q(x − 1), - x = 1, L-1, (1) with Q(0) 0 and Q(L) = 1. = d) Solve the difference equation in (1) analytically. Let P(x0) P{XT = 0}. Determine P(x) directly and show that P(x) + Q(x) = 1. Hint: Use the same form of the difference equation and analytical solution as in…Let the probability function of the random variable X bef(x) = { x/45 if x = 1, 2, 3, ⋯ ⋯ ,9 { 0 otherwiseFind E(X) and Var(X)
- Suppose that the length of time spent by a student with a professor during office hours in minutes is an exponential r.v. with parameter ?=1/10. This professor only sees students in the office hour one at a time. Thus, if you arrive to the office and there is someone inside, you have to wait. Consider 5 randomly chosen days in which you visit the professor's office during the office hour period and someone arrives immediately ahead of you. What is the probability that in one of those days you will have to wait between 10 and 20 minutes?The time (in hour) to failure of a component after starting a new machine is exponentially distributed with mean 3 hours. If 2 hours have passed with no component failure after starting the machine, what is the probability that the component will fail in the next hour? Choose the correct equation to aswer the question. Оа. Р(X 2) — Р(X 3|X 3|X > 2) = P(X > 1)Help me please
