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- Ук. Suppose that Y₁. Y₂; Yn 15. 2 from Function random Sample a population with probability density f(y) = find the maximum BY 1-B B like hood ozy 21: B >0 Elsewhere Estimator of B7. )The cumulative distribution function of a random variable X is defined by the equation if x1 a. Determine the probability density function of the random variable X b. Determine the probability random variable x from ½ to 1 c. Is the probability density function a valid PDF?4. Let Y U (0, 1). Show that X = -2ln(Y) follows an Exponential distribution with mean 2. pplications
- B4. Let X1, X2, ..., X, be a random sample from a normal distribution with known mean µ, unknown variance o? and density function 1 fx(x) IER. V2no2 Using the log-likelihood, find the maximum likelihood estimator for o?. If the mean is µ = 3 and the sample is {r1, 2, T3} = {-1.2,0.5, 1.5}, what is the estimate for o??Q5. A continuous r.v. X is said to have a Gamma (a, B) distribution if its moment generating function (m.g.f.) is given by M x (1) = (1- Bt). Let X1,X 2,.,X , be a random sample of size n from an exponential distribution with mean 1/0 and m.g.f. (1-0t). Let Y =EX; i =l 1 Ex [For p.d.f.s of Gamma and Exponential, and X i =1 (a) Find the m.g.f. My (t) of Y and state the distribution of Y specifying the parameters. (b) Find the m.g.f. M (t) of X and state the distribution of X specifying the parameters. (c) Suppose you have a random sample of size n = 100 from a Gamma (a, B) distribution with a = 3 and B = 5/3. Use the Central Limit Theorem to find P(4.5iid *1. Let Y1,..., Yn * uniform(0, 1). Find the distribution of their geometric mean: U = i=1 (Hint: consider first finding the distribution of –n log U.) - N (µ, o2). Find the distribution of Y and show that P(Y s e") = 0.5 (i.e., e" is *2. Let log Y the median of Y).Q1. Solve the following two problems for distributions:- a) Let X be uniformly distributed (continuous) on the interval [1,2]. Find E(1/X). b) Let us assume that the lifetime of light bulbs follows an exponential distribution with the pdf: f(x) = λ exp(-λx) We test 10 bulbs and their lifetimes are 5, 3, 5, 1, 7, 3, 4, 8, 2, 3 years, respectively. What is the MLE for λ? What is the method of moments estimator for λ?5. Let 0 and a be parameters. Show that the family of functions fo,a(x) = I > 0 (x+0)a+1³ is a probability density function. What can you say about the mean of a random variable that has fo,a as a pdf?2..You are given fT(30)(t) = t/1250 0≤t<50 0 o.w. a) Find the probability that (30) dies between ages 40 and 45. b) Find the probability that (60) dies between ages 65 and 85. c) Define the cumulative distribution function of T (45). d) Define the force of mortality of T (50).iid *1. Let Y₁,..., Yn uniform(0, 1). Find the distribution of their geometric mean: n -(II.) ΠΥ i=1 U = n (Hint: consider first finding the distribution of -n log U.) *2. Let log Y N (μ, 0²). Find the distribution of Y and show that P(Y ≤ eª) = 0.5 (i.e., eª is the median of Y).3. (a) A random variable, X, has the following probability density function: for 0 S<2 S(m) = (20 - 42)/30 for 2 S S6 otherwise. 1. Sketch the graph of f(#). (The sketch can be drawn on ordinary paper graph paper needed.) -no il. Derive the cumulative distribution function of X. ill. Find the mean and the standard deviation of X.2B. (a) An exponential distribution has probability density function ƒ(x) = ¹²e-x/a, x ≥0. α Show that the mean of the distribution is a and the variance is a². (b) A machine contains two belt drives, of different lengths. These have times to failure which are exponentially distributed, with mean times a and 2a. The machine will stop if either belt fails, and the failures of the belts are independent. Show that the chance that the machine is still operating after a time a from the start is e 32SEE MORE QUESTIONS