1. Tossing a DieSuccessive tossing of a die can be modeled as a stochastic process:X1, X2,: (S2, P) → {1, 2, 3, 4, 5, 6}.We assume that the random variables (Xt) teN are identically and independently distributed(iid).Problems to be solved in your lab session:wwwHIDHP.1: Explain in your own words why the assumption that all X+ are iid isjustified in the case of tossing a die.P.2: Give a real-world example of a stochastic process, where it is not justified toclaim that the random variables are iid.

Stochastic Process: A stochastic process is a family or set of random variables(ordered). The order…

Answered: 1. Tossing a Die Successive tossing of…

A First Course in Probability (10th Edition)

10th Edition

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Author:Sheldon Ross

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Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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3. Consider the random variables Y1,..., Yn that satisfy Y; = Bx; + €;, i = 1, ..., n. where x1,..., Xn are fixed constants, and e; are i.i.d. random variables that follow N(0, o²) where o2 is unknown. (a) Is B = EYi/Ex; an unbiased estimator of B? Show work. (b) Find the variances of 3 and ß. If you already found the variance of 3 from Problem 3 then you don't need to find it again. Which estimator has smaller variance? Which estimator is a better estimator? And why? Show work.
If X and Y are independent Poisson random variables with E[X] = 3 and E[Y] = 2, what is P[ X = 2 | X + Y = 4 ]?
Example 35: Is it possible to determine the random variable X for which Pm - 3 ox
..BERSAMA DI... 4G 3:18 PM Done Q2 a Sta Mid term.png Q2 (a) The discrete random variable has the probability distribution as in Table Q2. Table Q2: Probability distribution function X 0 1 3 4 5 P(X) 1 h 1 1 1 3 6 9 18 Find (11) Calculate the mean and variance. E(2X + 2) (iv) Var(3x + 1) 1 74% A
1. The random variables §, §1, §2,... are independent and identically distributed with 1/4 and P(§ = j) = c/j for j = 1,2,3. Let X₁ = 0 and Xn = max(§₁, ..., Èn) for n = 1, 2,.... distribution P(§ = 0) = (a) What value must c take? (b) Explain why {Xn, n = 0, 1, 2,...} is a Markov chain. (c) Write down the transition matrix. (d) Draw the transition diagram and classify the states (aperiodic, transient, re- current, eorgodic, etc). (e) Calculate P(Xn = 0). (f) Calculate P(X₁ = 3, X₂ = 1|X₁ = 3).
3. Students looked at the effect of a certain fertilizer on plant growth. The students tested this fertilizer on one group of plants (Group A) and did not give fertilizer to a second group (Group B). Let X and Y denote the respective growths of the plants (in mm) in Group A and Group B over six weeks. Suppose X and Y are independent random variables with distributions N(ux, o) and N(uy, o), respectively. A random sample from N(ux.o%) of size n = 25 yielded i 35.83 and s = 23.81, while a random %3! sample from N(uy o) of size m = 29 yielded y = 31.51 and s = 33.76. %3D (a) Assume o = o, test the null hypothesis at 1% significance level that the mean growths are equal against the alternative that the fertilizer enhanced growth. (b) If o #o, test the mull hypothesis at 1% significance level that the mean growths are equal against the alternative that the fertilizer enhanced growth. (c) Test Ho : o = oỷ against H1 : of + oỷ at the a = 0.05 significance level.
Aa.127. Let X1,X2,...,X4 be indepedent Poisson random variables with rates λ1,...,λ4, respectively. Show that X1 + X2 + X3 + X4 is a Poisson random variable with rate λ1 + λ2 + λ3 + λ4.
Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sample space. Suppose that we also have the following. E(Y)=-6 E(Z) =7 Var (X) = 31 Var(Y) = 34 Var(Z)=5 E(X)=2 Compute the values of the expressions below. E(5-5Y)= 0 (")-0 %3D Var(3Y+5) = 0 E(-x') - 0
10. Professor Snape is your instructor for Chemistry 101. He has given the same final exam for his class for a long time; he happens to know that exam scores on this test are normally distributed and have a population mean of µ=80. Because of prior laboratory catastrophes (don’t worry, all students were returned to their original species), he tried a new method of instruction this year. He got a random sample of n=16 students and taught them using virtual reality. These students then took his original final exam, and their sample average was M=76. --------unique information for the second question below: -------- Assume for this second question that Snape’s final exam has a population standard deviation of σ=6. Can he demonstrate that the virtual reality instruction method was significantly different from his traditional teaching method? (Assume a two-tailed test with α=.05) Yes, he can conclude that his VR method is significantly different from his traditional…
2b)
Could you help me with this problem about random variables?
Example 5-49. Let the p.d.f. of the random variable (x, y) be : a-² e-(x+y)/a ; x, y > 0, a > 0 f(x, y) = { , elsewhere Find the distribution of (X-Y).

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