Exercise 3 Let assume independent tosses of an unfair six-sided die as 1 1 random experiment with μ = 12'3'3'4'123) Compute likelihood of observing D = 2,4,5,1,3.
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- What is the probability that a SSRW on Z^2 starting at (0, 0) eventually hits the point (100, 100)?A coin weighted so that P ( H) =3/4 and P( T ) = 1/4 is tossed three times . Let X be the random variable which denotes the longest string of heads which occurs . The first noncentral moment of X isLet Mx, y be the moment generating function of random variables that are not independent of X and Y. Which of the following / which are not the properties of the function Mx, y?
- Three distinct integers are chosen at random from the first 20 positiveintegers. Compute the probability that: (a) their sum is even; (b) their product iseven.A single observation of a random variable havinga geometric distribution is to be used to test the null hypothesis that its parameter equals θ0 against the alter-native that it equals θ1 > θ0. Use the Neyman–Pearson lemma to find the best critical region of size α.Let X1 and X2 constitute a random sample from a nor-mal population with σ2 = 1. If the null hypothesis μ = μ0 is to be rejected in favor of the alternative hypothesis μ = μ1 > μ0 when x > μ0 + 1, what is the size of the criti-cal region?
- The combined amount of time three customers spend waiting to receive service at a drive-through facility is a Gamma distributed random variable with parameter theta = 0.4 per minute. What is the probability that 3 randomly selected customers will have a combined waiting time of at least 10 minutes?a data managment company records a large amount of data. Historically, there are 1.5 errors per 100 pages of data. Let x be the random variable representing the number of errors in 100 pages. i) if X has a poisson distributiuon, write down value of its paramaters ii) what is the probability that there are three errors in a randomly selected 100 pages? ii) what is the probability that there are three or more errors in a randomly selected 200 pages?Suppose that random variables X and Y are defined on a sample space with only two elements. Suppose that Cov(X, Y ) = 0. Prove that X and Y are independent.
- A bag consists of 11 blue coins and 12 red coins. Three coins are selected from the bag in order. If the selected three must contain at least one red coin then how many different possibilites of selecting are there?If X, Y are normally distributed independent random variables, then show that W = 2X - Y is normally distributed. Do not use moment generating function, only use convolution formula.A round spinner is divided into five sections, where the sections do not have the same size. Using the measure of the interior angles, the probability of the spinner landing on any individual space on a spin is calculated and given in the table. Section 1 2 3 4 5 Probability 0.40 0.20 0.20 0.15 0.05 A statistics student is asked to test the integrity of the spinner using a chi-square goodness-of-fit test. What is the minimum number of times the spinner should be spun to conduct this test? A. 5 B. 25 C. 50 D. 100 E. 500