Q.2 (а) Three marbles are drawn without replacement form an urn containing 4 red and 6 white marbles. Let X be a random variable that denotes the number of red marbles drawn. i) Construct a table showing the probability distribution of X ii) Find P(X=2), P(1< X < 3), and interpret the results.
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Q.2 (а) Three marbles are drawn without replacement form an urn containing 4 red and 6 white marbles. Let X be a random variable that denotes the number of red marbles drawn.
i) Construct a table showing the probability distribution of X ii) Find P(X=2), P(1< X < 3), and interpret the results. A continuous X = 2 and X (a) can assume values between om 5has a density
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- 4. Suppose that discrete random variable RV take values {1, 2, 3, 4,.,k,.} with probability 1/2. a) What is its Cumulative Density Function (CDF)? b) Estimate the Variance of these data sets.A probability distribution function is modeled by the following X-Y cartesian graph: 15k 长 6 7 12 Determine the following: 1. Probability density function for the intervals defined on the graph. 2. Expected value of X. 3. Variance of X. I 4. P(X<5) 5. P(2For a statistic to be a good estimator of a parameter, two properties it must satisfy are unbiasedness and minimum variance. Consider a sample of three observations X₁, X₂, X, where X, ~ Exp (0). That is, a sample of size 3 is taken from a population 2 3 following the exponential distribution with density function given by f(x) = 1e % if x > 0 0, Otherwise. Five possible estimators of are â‚ =X₁, Ô₂ = ¹/(X₁ + X₂), Ô‚ = =— (X₁ + 2X₂), Ô¸ = X, and Ô¸ = ¹⁄ (X₂ + X). 2 [Hint: Use the fact that for variable X we have E(X)= 0 and E(X²)=20² and Var(X) = 0². (a) Show that the five estimators given above are unbiased for 0. (b) Find the variances of each of the five estimators. (c) Which estimator will you choose for 0. Why?Let X and Y be independent Exp(2) random variables. Define W = X + Y. a) Determine the correlation between X and W. The joint pdf of X and W is: fx,w(x, w) = 1^(2) e^(-1w) I{Ob) Let X₁, X₂,..., X, denote a random sample of size n from a distribution with probability density function: i) ii) iii) f(x; 0) = {0(1-x)(¹+0), Find the maximum likelihood estimator of 0. Derive fisher's information of 0. suppose that 5 independent observations were taken from such a distribution: 4 1 3 5 4 and 0 = 0.7052 Find a 99% confidence interval for 0. x>0 elsewhere Da. Explain the difference between the method of moment generating function and the method of incarnation to find the probability density function of a random variable b. Let X1, X2, X3 ,,,,, each of them has Poisson distribution with parameter lamda 1, lamda2, lamda3,........ i. Get the probability distribution of Y=X1+X2+....Xn ii. Get the mean and variance of Y c. Y1 and Y2 be a joint probability function as shown below in the screenshot imageU=Y1/(Y1+Y2), Using the method of incarnation (Jacobian), find the probability density function of U1 A continuous random variable is uniformly distributed between a and b. The probability density function between a and b is a b (а- b) (b - a) d 1/(b - a) 2 The assembly time for a product is uniformly distributed between 6 and 10 minutes. The probability of assembling the product between 7 and 9 minutes is a b 0.5 0.2 1 3 Larger values of the standard deviation result in a normal curve that is shifted to the right a b shifted to the left narrower and more peaked wider and flatter d 4 For the standard normal probability distribution, the area to the left of the mean is a -0.5 b 0.5 any value between 0 and 1 d 1 5 Assume z is a standard normal random variable. Then P(1.41Let Y1, Y2,..., Y, denote a random sample from the density function given by 1 yª-'e=y/®, y> 0, f(y[a, 0) = elsewhere, where a > 0 is known. a Find the MLE Ô of 0. b Find the expected value and variance of ê. c. Is the MLE ô an unbiased estimator for 0?If a discrete random variable X has the following probability distribution: X -2, - 1, 0, 1, 2 P(X) 0.2, 0.3, 0.15, 0.2, 0.15 Use this to find the following: (a) The mean of X and E[X^2]. (b) The probability distribution for Y = 2X^2 + 2 (i.e, all values of Y and P(Y )). (c) Using part (b) (i.e, the probability distribution forY ), find E[Y ]. (d) Using part (a), verify your answer in part (c) for E[Y ]. **Note: Please do not just copy from Chegg!Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.Actuary Tong has to study for two actuarial exams: Exam P and Exam FM. The amount of study time that Actuary Tong will spend on each exam in a day follows a continuous random variable that ranges from o to i hour. The amount of study time that Actuary Tong spends on both exams in a day has a joint density function that is equal to the sum of the study times that Actuary Tong spends on ench exam in a day. Caleulate the probability Actuary Tong spends at least half an hour in a day studying for exactly one of the exams. A 0.45 B 0.50 0.55 D 0.60 0.65 REPRecommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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