(a) Find the MLE for 0, say . Be sure to check the second derivative. (b) Find the method of moments estimator of 0, say ē. 1000 (c) Suppose that y₁ = 750.7516 and 1000 vi=1 10 In(yi) = -329.0056, calculate and 0. i=1
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![2. Let Y₁, Y2,..., Yn denote a random sample from the probability density function
(0+1)yº; 0<y< 1, and 0 > −1
otherwise.
f(y;0) = {
0,
(a) Find the MLE for 0, say . Be sure to check the second derivative.
(b) Find the method of moments estimator of 0, say .
(c) Suppose that 1000 'yi = 750.7516 and 1000In(y) = -329.0056, calculate 8 and 0.
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- 2. Let X be a continuous random variable with probability density function (pdf) 3x�� 0* Consider the probability density function fx (x) = a e-b lel where X is the random %3D variable which assumes all the values from (i) relation between a and b -00 to -00, Find (ii) the probability of finding X in the range 1 to 2.4. Let 0 be a parameter. Show that the family of functions Oe-0/x fo(x) = I > 0 is a probability density function. What can you say about the mean of a random variable that has fo as a pdf?A variable X is distributed at random between the values 0 and 4 and its probability density function is given by: f(x) = kx³ (4- x)². Find the value of k, the mean and standard deviation of the distribution.A continous random variable has density function f(x)=(3x?)/(63) for Osxse, for some 0>0, and zero otherwise. What is the ratio of the mode to the median for this distribution? A. 2(1/3) B. 0/(21/3) O C.e D. 2(-1/3)Figure I shows the piecewise function (I), (II), (III) and (IV) for cumulative distribution function F(x) for continuous random variable. F(x) (6. 1) IV III II (4.0.8333) (0.0.1667) Figure I Construct the probability density function fix). Should one of the piecewise functions (IV) is not constant, explain the changes.1. Let f be the probability density function of a continuous random variable X whose range is (-0, 00), where if a sxsb f(x) ={32 0 otherwise. (a) Explain why a, hence b, must be nonnegative. Graph the function f. (b) Find the value of b'-a. 7 (c) If it is known that P(X 8).10) Let f(x) by the probability density function of some continuous random variable. Explain why it must be true that f(x) dx =1.3. The random variable a represents the annual rainfall depth in El Paso in inches. Be- tween values of r = 0 and r = 15, the probability density function has the equation 1 f(x) = 15 07. )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?Let X be a random variable with pdf f(x) = kx*,-16. Let Yi and Y2 be independent exponential random variables, both with mean 0>0. Let U = - and V = Y, +Y,. Using a change of variable technique, %3D Y, +Y; (a) find the probability density function of U. (b) find the probability density function of V.SEE MORE QUESTIONSRecommended textbooks for youCalculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,Calculus For The Life SciencesCalculusISBN:9780321964038Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.Publisher:Pearson Addison Wesley,