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- 3. You are given that S60 (t) = 2 – e/90 for t < 62 a) What is the survival function for a person who is now 80 years old? b) What is 10|2470 ? c) State in words what probability the previous question is asking for.Sn/2", l1/2". if n is a prime number otherwise. Let an Does Ea, converge? Give reasons for your answer.1. Let X ~ Poisson(A) and Y ~ Poisson(u). Assume that X and Y are independent. Use probability generating functions to find the distribu- tion of X + Y.
- A political poll is taken to determine the fraction p of the population that would support a referendum requiring all citizens to be fluent in the language of probability and statistics. (a) Assume p = 0.5. Use the central limit theorem to estimate the probability that in a poll of 25 people, at least 14 people support the referendum. Your answer to this problem should be a decimal. (b) With p unknown and n the number of random people polled, let Xn be the fraction of the polled people who support the referendum. What is the smallest sample size n in order to have a 90% confidence that Xn is within 0.01 of the true value of p? Your answer to this problem should be an integer.The probability that machine A will be perform- ing a usual function in 5 years’ time is 1/4 while the probability that machine B will still be operating usefully at the end of same period is 1/3. Find the probability that in five years' time : (a) both machines will be performing a usual function, (b) neither will be operating, (c) only machine B will be operating, and (d) at least one of the machines will be operating.Suppose X-U (-54, 60) and F(t) is the cumulative distribution function. What is the probability that X is in the interval [-51, -21] or in the interval [-36, 57]? O (F(-21)-F(-51))+ (F(57) - F(-36)) O (F(-21)-F(-51)) x (F(57) - F(-36)) O F(-21)-F(-36) O F(57) - F(-51)
- Let {Xn}-1 be a sequence of random variables that converges in probability n=1 Prove that the limiting random variable is unique P-a.s., i.e., if Xn 4 X and Xn 4 Y, then P(X = Y) = 1.For the M/M/N/o system, the probability that an arrival will find all servers busy and will be forced to wait in queue is an important measure of performance of the M/M/N/∞ system. This probability is given by PQ N!(1 – p/N) | and is known as the Erlang C formula. Please derive the equation. What is the expected number of customers waiting in the queue (not in service)?The duration, x, of a monthly faculty meeting is uniformly distributed between 30 and 80 minutes. Which of the following statements is true (check all that apply)? a. Prob (x>80) = 0 b. Prob (x<85) = 1 c. Prob (x>35) = 1 d. Prob (x<30) < 0