Question:

(Cauchy dissemination.) An arbitrary variable X with thickness ? co 0

1 a f(x) = 1r a2 ?/7)2

is said to have a Cauchy dissemination with boundaries an and b. This dispersion is fascinating in that it gives an illustration of a constant irregular variable whose mean doesn't exist. Let a = 1 and b = 0 to acquire an exceptional instance of the Cauchy conveyance with thickness 1 f(x) = 1 + x= - co

Question 10

Let Xdenote the measure of time in hours that a battery on a sunlight based adding machine will work enough between openings to light adequate to re-energize the battery. Accept that the thickness for X is given by .f(x) = (5016)x-3 2

(b) Discover the articulation for the aggregate circulation work for X, and use it to discover the likelihood that an arbitrarily chosen sunlight based battery will last all things considered 4 hours prior to waiting be re-energized.

(c) Figure out the normal time that a battery will last prior to waiting be re-energized.

(d) Discover E[X2], and utilize this to discover the fluctuation of X.

Question 11

Accept that the increment sought after for electric force in great many kilowatt hours throughout the following 2 years in a specific zone is an irregular variable whose thickness is given by KO= (1/64),0 0

(b) Discover the articulation for the total dispersion for X, and use it to discover the likelihood that the interest will be all things considered 2 million kilowatt hours.

(c) If the territory just has the ability to create extra 3 million kilowatt hours, what is the likelihood that request will surpass supply?

(d) Track down the normal expansion sought after.

Question 12

The mass thickness of soil is characterized as the mass of dry solids per unit mass volume. A high mass thickness suggests a minimized soil with not many pores. Mass thickness is a significant factor in affecting root advancement, seedling development, and air circulation. Allow X to mean the mass thickness of Pima earth soil. Studies show that X is typically dispersed with p = 1.5 and a = .2 g/cm3. (a) What is the thickness for X? Sketch a chart of the thickness work. Show on this chart the likelihood that X lies somewhere in the range of 1.1 and 1.9. Discover this likelihood.

(b) Discover the likelihood that a haphazardly chosen test of Pima earth soil will have mass thickness under .9 g/cm3.

(c) Would you be shocked if an arbitrarily chosen test of this sort of soil has a mass thickness in overabundance of 2.0 g/cm3? Clarify, in view of the likelihood of this happening.

(d) What point has the property that lone 10% of the dirt examples have mass thickness this high or higher?

(e) What is the second creating capacity for .X?

6. A Markov chain with the states 1, 2, 3 has transition probability matrix: 1 . P: 2 0.4 0.3 b 3 (a) Give the values of a, b, and c that make the above a valid transition probability matrix. (3 points) (a) Define the properties of a homogeneous Poisson process. Include a conceptual diagram illustrating the process, and label the diagram to support your answer. (b) How is an inhomogeneous Poisson process different? (c) Describe how the following distributions are related to a homogeneous Poisson process with rate A. [3 marks] Poisson (X) f(x) = exp d Exponential(A) f(x) = Aexp-At 120 Gamma(n, A) f(x) = Aexp-At (At)-1 (n-1)! 120 Your description should identify what each distribution is used to model.(ii) if the Xx are i.i.d. RVs, then os_ (s) = ox, (s). (8) Problem 8. (Moment generating function of the Poisson process) (i) Determine the MGF (problem 5) of the Poisson process Me with rate A(t) from knowledge of the count distribution PA(t) -MOA(t) where A(t) = )= Xu)du. (9) (ii) Then use the properties of the MGF listed in problem 7 to show that the sum of independent Poisson processes is a Poisson process. Remark: Note that for a stochastic process, the MGF is a function of both s and f, in the same say as the PMF 9 is a function of both & and f. Matlab projecthttp://keisan.casio.com/exec/system/1180573226 Unlike the normal distribution which is defined over the entire x-axis (that is, under the normal distribution there is no theoretical minimum or maximum the random variable x can take), the beta distribution has a "finite support" (that is, the beta pdf is only defined for x in the interval [0,1]), and it can be use use to define similar "finite support" distributions over any given interval. This means that if you are studying a random variable whose values are bounded above and below, you may be able to fit its histogram using some derivation of the beta distribution. The beta distribution has two "shape" parameters a and b. The following experiments will help you understand how the shape parameters a and b work. FYI, here's an explanation https://www.youtube.com/watch?v=v1uUgTelnQk 1. What is the shape of the beta distribution if a = 0.5 and b = 0.5? 2. What is the shape of the beta distribution if a = 0.5 and b = 1? 3. What is the shape of the beta distribution if a = 1 and b = 1? 4. What is the shape of the beta distribution if a = 3 and b = 3? 5. What is the shape of the beta distribution if a = 30 and b = 30? 6. What is the shape of the beta distribution if a = 300 and b = 300? 7. What is the shape of the beta distribution if a = 3 and b = 20? 8. Find a and b such that the resulting beta distribution is skewed right. 9. On a spreadsheet create a list of forty numbers between 0 and 1 whose histogram follows the shape of the distribution in part 8. (Hint: just as when you generated a perfectly normal distribution a few weeks ago using the normal standard inverse function, here do the same thing; but this time use the beta inverse function "beta.inv(x,a,b)" where x will be the values you get after you divided the interval into 40 equal sized pieces.) 10. Now apply a transformation so that your values are centered around 100 with variance 100; in other words they have mean = 100, and standard deviation =10