7.80. (Stationary Distribution.) If (π;) are stationary probabilities for (X+), thenwe have seen that \To = μπ₁, and for j ≥ 2,=(λj + μj)πj = λi−1πi−1 + µj+1πj+1.(7.41)Show that Tk=λολη-1με... με•ΠΟ If C =1 + x=1λολη-1με...μη< ∞, show that thestationary distribution is given by ñ = 1/C and(7.42)πn =1 λοC με ...μηAn-1n ≥ 1.'

the equation in the image you sent because it relies on specific details about a stationary…

Answered: 7.80. (Stationary Distribution.) If…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Consider two securities, the first having μ1 = 1 and σ1 = 0.1, and the secondhaving μ2 = 0.8 and σ2 = 0.12. Suppose that they are negatively correlated,with ρ = −0.8. Denote the expected return and its standard deviation as functions of π byμ(π ) and σ (π ). The pair (μ(π ), σ (π )) trace out a curve in the plane as πvaries from 0 to 1. Plot this curve in R.
Suppose that you have ten lightbulbs, that the lifetime ofeach is independent of all the other lifetimes, and that eachlifetime has an exponential distribution with parameter l.a. What is the probability that all ten bulbs fail beforetime t?b. What is the probability that exactly k of the ten bulbsfail before time t?c. Suppose that nine of the bulbs have lifetimes that areexponentially distributed with parameter l and thatthe remaining bulb has a lifetime that is exponentiallydistributed with parameter u (it is made byanother manufacturer). What is the probability thatexactly five of the ten bulbs fail before time t?
Suppose the lengths of human pregnancies are normally distributed with u = 266 days and 0 = 16 days. The area to the left of X = 245 is 0.0947. What is the probability that a randomly selected human pregnancy lasts
Suppose that X has a Poisson distribution with h = 80. Find theprobability P(x is less than or equal to 80)
Suppose that a random variable X has a hypergeometric distribution with parameters n = 36, N = 50, and M = 28. What is the mean of X? Do not round!
2. Suppose that at a gas station the daily demand for regular gasoline is normally distributed with a mean of 1000 gallons and a standard deviation of 100 gallons. The station manager has just opened the station for business and there is exactly 2300 gallons of regular gasoline in storage. The next delivery is scheduled later tomorrow at the close of business. (a) Let X1 and X2 denote the demand of today and tomorrow, respectively. Let X = X1 + X2. Find the distribution of X. (b) Find the probability that the manager will have enough regular gasoline to satisfy today’s and tomorrow’s demand.
Show that, if X1 and X2 are independent random variables with geometric distributions of aparameter p, then Y = X1 + X2 has a negative binomial distribution with parameters p andk = 2.
A masking tape manufacturer expects 0.04 flaws per meter of tape, on average. The Poisson assumptions hold. Find the probability of exactly 1 flaw in 0.1 meters of tape. 0.0138 0.0289 0.0103 0.0221 0.0040
1) A producer of soft drinks was fairly certain that her brand had a 15% share of the soft drink market. In a market survey involving 3000 consumer of soft drinks, 250 expressed a preference for her brand. If the 15% figure is correct, find the probability of observing 400 or more consumers who prefer her brand of soft drink. [Hint: Use normal approximation and z-value must be rounded-off to 2 decimal places only
Suppose the distribution of the time $X$ (in hours) spent by students at a certain university on a particular project is gamma with parameters $\alpha=50$ and $\beta=2 .$ Because $\alpha$ is large, it can be shown that $X$ has approximately a normal distribution. Use this fact to compute the approximate probability that a randomly selected student spends at most 125 hours on the project.
In the daily production of a certain kind of rope, the number of defects per foot given by Y is assumed to have a Poisson distribution with mean ? = 4. The profit per foot when the rope is sold is given by X, where X = 70 − 3Y − Y2. Find the expected profit per foot.
Suppose you take independent random samples from populations with means μ1 and μ2. Furthermore, assume either that (i) both populations have normal distributions, or (ii) the sample sizes (n1 and n2) are large. If X1 and X2 are the random sample means, and S1 and S2 are the random sample standard deviations, then how does the quantity behave? Give the name of the distribution and any parameters needed to describe it.

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