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- A random sample of 50 suspension helmets used by motorcycle riders and automobile race-car drivers was subjected to an impact test, and on 18 of these helmets some damage was observed.2. We have n = 100 many random variables X¿'s, where the X¿'s are independent and identically distributed Bernoulli random variables with p = 0.5 (E(X;)=p and var(X;)=p(1-p)). n a. What distribution does X, follow exactly (sum of independent Bernoulli random variables)? State the type of distribution and what the parameter is. i=1 = b. Using the central limit theorem, what approximate distribution does X the type and the mean and variance. answer. HƏ IO SC = EX, is greater than 0.5? Write down the c. What is the approximate probability that X integral using proper notation and use empirical rule or pnorm function in R to obtain numeric i=1 n n n EX, follow? State i=1Let Yı, Y2,., Yn be a random sample from the following distribution fy) = {02 for 0sy 202 elsewhere With parameter 0>0. Prove that E1 y? is a sufficient statistic for the parameter 0.
- Q10. Let X1,X 2,.,X, be a random sample of size n from a distribution with mean 0 and variance o. Consider the following two estimators for 0: Ô, = X, and Ô, = X. (a) Show that Ôj and Ô2 are both unbiased estimators of e. (b) Obtain the relative efficiency of Ô to Ô2. (c) Which estimator, Oj or O2 is a more efficient estimator? Justify your answer.6.. Let X1, X2, ., X, denote a random sample from a Bernoulli distribution with parameter p. Find a sufficient statistic for p.4
- 1. Let X1,X,....,X, be random sample from Geometric distribution a a fx(x) = p(1– p)*' , x=1,2,3,... a. F,(y,) =? b. F, (yn)=? с. Р(\ <1)-?8. Let X1, X2 be i.i.d. random variables from a distribution with p.d.f fx(x) = otherwise and let Y = X(1) = min(X1, X2). Calculate P(1Let Y1, Y2, ... Yn be n independent random variables. E(Yi) = μ and Var(Yi) = o² for every 1 = 1,2, ... n (NOTE: W is the sample mean) Let W = 1 yi n n (a) (b) Find E(W). Find Var(W).A random variable Y has a uniform distribution over the interval (8,, e,). Derive the variance of Y. Find E(Y)? in terms of (e,, e,). E(Y)? = Find E(Y) in terms of (e,, e,). E(Y) = Find V(Y) in terms of (8,, e,). V(Y) =The independent random variables X1, X2, X3, X36 all have mean µx = 10 and standard deviation ... Ox = 3. Find the probability that their mean is in the interval (9, 11), i.e. find P(9The Insurance Institute for Highway Saftey publishes data on the total damage suffered by compact automobiles in a series of controlled, low-speed collisions. A sample of the data in dollars, with brand names, removed, is 950,650,850,750 The interquartile range of the above data is?SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON