Answered: Suppose that {uj. t… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.2: Arithmetic Sequences

Problem 68E

See similar textbooks

Related questions

Question

Suppose that {u¡ t = ...,-2,-1,0,1,2,...} is an independent time series with mean 0
variance o = 4.0. Suppose that the time series {xj: t = ...,-2,-1,0,1,2,...} satisfies the
equation.:
x=0.5 x_₁ + 1.0+ u, -0.4 -1.
Determine the
1. mean,
2. autocovariance function,
3. variance,
4. autocorrelation function,
5. Partial autocorrelation function of the time series x₁.

Expert Solution

Step by step

Solved in 7 steps with 68 images

SEE SOLUTION Check out a sample Q&A here

Blurred answer

Similar questions

Suppose that {u: t = ...,-2,-1,0,1,2,...} is an independent time series with mean zero, variance σ²= 9.0. Suppose that the time series {x+: t = ...,-2,-1,0,1,2,...} satisfies the equation: 2 X= 1.5 X-1 - .5 X-2 +3.0+u- U-1- 3 a) Identify the time series. b) c) d) i) Determine the autocovariance function and autocorrelation function of the time series X+ - Xt-1- Find the random shock form of the time series. Suppose that the first five observations of the time series are x₁ = 2.5, x2 = 4.0, X3 = 3.7, x4 =1.0 and x5 1.5. = Use these observations to compute prediction intervals for the next 5 observations. (Compute both 95% and 66.7% prediction limits) ii) If the sixth observation turns out to be x6 = 4.3 use this information to re-compute prediction intervals for the next 4 observations. (Compute both 95% and 66.7% prediction limits)
A system consists of five identical components connected in series asshown: ----1-----2-----3-----4-----5---- As soon as one component fails, the entire system will fail. Suppose eachcomponent has a lifetime that is exponentially distributed with λ= .01and that components fail independently of one another. Define events Ai [ith component lasts at least thours], i=1,..., 5, so that the As areindependent events. Let X= the time at which the system fails-that is,the shortest (minimum) lifetime among the five componentsa. The event {X≥t) is equivalent to what event involving A1,..., A5? b. Using the independence of the Ai's, compute P(X≥t). Then obtain F(t)= P(X≥ t) and the pdf of X. What type of distribution does Xhave?c. Suppose there are n components, each having exponentiallifetimewith parameter 1. What type of distributiondoes Xhave?
Suppose that a time series data follows an MA(1) model, calculate cov(Xt, Xt-p) for all p≥ 1.
Thank you
•Suppose xau (-54,60) and F(t) is the Cumulative distribution function. which is the probability that x is in the interval E-51.-21] and in the interval [36,57] A. F(57)-F(51) B.(F(-21)- F($1)) +(F(57)-F(-36)) C. (F (-21)-F(-51) × (F(57)-F(-36)) D. F(-21) - F(-36) (6)
Suppose that a time series process {y,} is generated by y, = z + er, for all t = 1, 2, ..., where {e;} is an i.i.d. sequence with mean zero and variance o?. The random variable z does not change over time; it has mean zero and variance o. Assume that each e, is uncorrelated with z. (i) Find the expected value and variance of y,. Do your answers depend on f? (ii) Find Cov(y, Ye+a) for any f and h. Is {y,} covariance stationary? (iii) Use parts (i) and (ii) to show that Corr(y,, Yr+k) = o(o + o:) for all t and h. (iv) Does y; satisfy the intuitive requirement for being asymptotically uncorrelated? Explain.
, Let X1, X2, ..., X500 be a white noise time series of length T = 500 following N(0, 1). Find the covariances Cov(X10, X11), Cov(X10, X9) and Cov(X10, (X9 + X10)).
A system consists of five identical components connected in series as shown: T-2H 3- As soon as one components fails, the entire system will fail. Suppose each component has a lifetime that is exponentially distributed with 1 = 0.01 and that components fail independently of one another. Define events A, = {ith component lasts at least t hours}, i = 1, . .., 5, so that the As are independent events. Let x = the time at which the system fails-that is, the shortest (minimum) lifetime among the five components. (a) The event {X 2 t} is equivalent to what event involving A,, O A, U A2 U A3U AqU A5 O A, U A, NA, U A NA5 O A, N A, U A3 NAUAS O A, N A, N Az n ANA5 (b) Using the independence of the A,'s, compute P(X > t). P(X 2 t) = Obtain F(t) = P(X < t). F(t) = Obtain the pdf of X. f(t) = What type of distribution does X have? O xis a gamma distribution with parameters a = 0 and B = 1. O xis an exponential distribution with 2 = 0.05. O xis a gamma distribution with parameters a = 1 and B =…
Suppose X1, X2, ... , Xn is a random sample and Xi = {1, with probability p 0, with probability 1-p} for every i = 1, 2, ... , n. Find the Moment Generating Function of ∑i=1n Xi . What is the distribution of ∑i=1n Xi ?
If two events A and B are such that P(A) = 0.3; P(B) = 0.4 and P(A' B') = 0.5, then find the value of P(B/(AU B')).
Let rt be a log return. Suppose that r0, r1, . . . are i.i.d. N(0, 0.01^2).(a) What is the distribution of rt(8) = rt + rt−1 + rt−2 +...+ rt−7?(b) What is the covariance between r7(3) and r9(3)?(c) What is the conditional distribution r17(3) given that r16 =0.004 (d) What is the probability that the gross return over the first 10 times periods is at least 1.05?
Show that the average time till the first head is observed is y₁= p-¹, where p is the probability of getting a head each time the coin is tossed. Then find a closed form explicit solution to yn, the average time till n consecutive heads are observed for the first time.

SEE MORE QUESTIONS

Recommended textbooks for you

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

SEE MORE TEXTBOOKS