3. Given the cumulative distribution function of a...

3. Given the cumulative distribution function of a continuous variable X, F(x) = { 0, x < 0 Ax^2, 0 ? x < 1 1, 1 ? x Please solve the following questions. (1) Constant A; (2) Probability of event when X is in (-1, 1/2); (3) Density function f(x). 4. Please solve the moment estimation ?? adopted in the distribution of variable X, which has a density function given below. X1, X2, …, Xn are samples. f(x) = { (? + 1)x^?, 0 < x < 1 0, otherwise

Transcript

00:01 Hi, i'm david and i'm here to have your answer your question.

00:03 Now in the question we have the two questions three and the four.

00:07 And let me do the question three first for you.

00:10 In the first question three, we're given the cdf on the random rumble x equal to the 0 if the x smaller than 0 equal to a x square if the x between the value of the 0 and 1 equal to 1 if x squared than equal to 1.

00:28 Now in the first part, a, we need to find the constant a.

00:34 To find the constant a, we know that we will, if we're blocking the x equal to one inside, this one must exactly equal to one.

00:43 And then if you put the one inside this function, then we have a times the x square, and then we get this one will equal to, this is implies that the a must, it will equal to, it will equal to the one square and we have the one square it will equal to the one so doesn't implies that the a must equal to the one and that's going to be the answer for the a now for the b once you find the probability of the x here it will be between the minus one to the one half now because this f x only valid between zone one so this is be the same as the probability of the x between the zoh and the one half.

01:34 And then for this one, one half will be inside this interval.

01:40 Then it should equal to the half of the one half.

01:45 And then we get equal to the i will find one already.

01:49 So we have one times under one half square.

01:53 And then we get equal to one over four and equal to the zubon 25.

01:58 Now for the question c, once you find the pdf, we know that the pdf can equal to the derivative of the cdf and then we can get derivative of this one, i equal to 1, so we have the 2x and for the x between the interval from zone 1 and equal to 0 otherwise.

02:22 And that's going to be the answer for the question 3.

02:25 Now for the question 4, we are given the dancer.

02:32 Fx equal to the theta plus 1 and then x to the power theta for the x in the interval 0 to 1.

02:43 0 otherwise.

02:47 And the question asks me to find the moment estimator theta heart...

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