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- An ordinary (fair) coin is tossed 3 times. Outcomes are thus triple of “heads” (h) and tails (t) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of tails in each outcome. For example, if the outcome is hht, then R (hht)=1. Suppose that the random variable X is defined in terms of R as follows X=6R-2R^2-1. The values of X are given in the table below. A) Calculate the values of the probability distribution function of X, i.e. the function Px. First, fill in the first row with the values X. Then fill in the appropriate probability in the second row.Let X be a random variable with the function, -(x -A) "for x > 2, Let f (x) = { elsewhere. Derive u and show that T = X is biased estimator of 2 . Can you modify T =X in order to get an unbiased estimator of 2.Suppose X~ Binom(6, p) and define an estimator T ==to estimate p. Then, variance 6. of T is, p(1–p) a) Var (T)= 6. b) Var (T) = 6 p(1 – p) c) Var (T)~ p(1-p) d) Var (T) - 36р
- Let X1,…,Xn be i.i.d. Exp(λ) random variables, where λ is unknown.What is the variance and quadratic risk of the unbiased estimator θ^ = n*mini(Xi).1) Find the MGF of the random variable X, the pdf is given by x E [0,1) ,x E [1,5) ,otherwise f(x) = }5-x Hence, find its mean and variance.Suppose X1, X2, . ,X, be iid random samples from Uniform(a, b), where b > a. ... Find the method of moments estimators for a and b and find the MLES of a and b.
- 3. The length of time required by students to complete a 1 hour exam is a random variable with a pdf given by: f (x) = = ca + for 0 sæ<1 a. Find c. Enter c as a reduced fraction. C = b. Find F(x). Enter coefficients as reduced fractions and use ^ to denote powers. F(x) = c. Find the probability that a student takes less than 30 minutes to complete the exam. Enter the probability as a reduced fraction. prob = d. Find the median length of time to take the exam. Enter your answer in hours with 4 decimal places. median = hours e. Find the length of time for the first 10% of the students to complete the exam. Enter your answer in hours with 4 decimal places. hours answer = f. Find the expected value, variance, and standard deviation of X. Enter your answer in hours with 4 decimal places (or hours squared in the case of variance), hours expected value =Suppose that X₁, the number of particles emitted in t hours from a radio - active source has a Poisson distribution with parameter 20t. What is the probability that exactly 5 particles are emitted during a 15 minute period?Suppose x binom(6, p) and define estimator T=x/6to estimate p. Then, variance of t is
