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(10 points) Based On random sample o size from the pdf below derive the MLE of 0.f (~;0) (0 F I)"<r<1, 0 € (-1,0)...

Question

(10 points) Based On random sample o size from the pdf below derive the MLE of 0.f (~;0) (0 F I)"<r<1, 0 € (-1,0)

(10 points) Based On random sample o size from the pdf below derive the MLE of 0. f (~;0) (0 F I)" <r<1, 0 € (-1,0)



Answers

Find the covariance of random variables $X$ and $Y$ having the joint probability density function $$ f(x, y)=\left\{\begin{array}{ll} x+y, & 0<x<1,0<y<1 \\ 0, & \text { elsewhere } \end{array}\right. $$

Problem 13. We want to identify this probability density function, then find the mean variance and the standard division. Without integration we can rewrite F of X To Equal one divided by 10 0. This is in the fourth one divided by b minus A. Because B equals 10 And the equal zero, then we can see that this probability density function is unified. And for a uniform distribution new equals half, multiplied by a plus B equals half, multiplied by a zero plus beaten Equals half, multiplied by 10 caps. five. For the variants it equals one divided by 12, multiplied by B minus a old square equals one bite by 12. But buoyed by B 10. A zero zero old square. Then it's 100 divided by 12 gives 25, divided by three. Finally, for a standard division it equals the square root of the variance, equals five by by Square Root of three. Or in decimal equals 2.8. Yeah. Seven.

Hi here it is given. In the first problem that we have in random samples explore next to extent which follows the distribution. Each follows distribution of the pdf F. X. Data. It is our parameter of interest will find that Amelia peter maximum likelihood estimator of theater. So for that we first find the likelihood function. Okay now it is given that the first problem affects the data provided density function is Theta Extra The World Theatre -1. zero less. The next less than one and three to get at 10:00 and zero. Otherwise. Okay now the likelihood function of theta even excalibur denoted by L. Theta colon. Ex girl that is Productive Iran's from 1 to N. If excite to to so we'll get it seated to the bird. End Product of islands from one doing excite all the power. The domain is What now we find that. Hello of theater says that this likelihood function L theater is maximum. Okay no maximizing this likelihood function is equivalent to maximizing log likelihood function because log function is increasing function of X log it is increasing function of X. Now we'll consider it log likelihood function equals to script L. Theater X. Skull which is a gold do in la vida plus 3 to -1 summation log extinct islands in one I have to find out in a similar instrument. Will you first find the first order derivative of log likelihood uh log likelihood function then acquitted. 20. Okay now one start a devotee of script. L. T to Eggs girl equals to zero. So we get in my theater plus submission dog exciting. I don't want any calls to zero. So from that equation we have theta equals two minus in whole liberated by summation. No Exciting items from one. This is today's speed ahead. Is that the head the estimate value of Peter now we again fine. The second order derivative evolved look like lewd function. Certainly regular function. What should I go to the diet? If it is negative then you can say that for the tigers to keep ahead the likelihood function or the log likelihood function Adams the maximum value and that is the family of theater. So find it this coming maximum or not. So second or identity will be minus in my theater square for the day college to get ahead. We have my nurse submission log. Excite gold square all divided by N. C. This time is in square. So definitely it will be getting the records to zero and again N. N definitely in desert and positive value. So the whole town will be negative. Right? So for Chicago should be died. It added the maximum value. So for that the Emily's Peter had Emily Equals 2- in Holy weighted by Submission Islands from one plane lock Xing. Mhm. This is the answer. Yeah. Going forward. The second problem we have the provided into the same X one X. Two accent These random temples follow the distribution with the B. D. F. F. X. Theater. Now Where FX. three days given It really were -X -3 to does that theater less than equals two X. Less than infinity. And theater better than zero and zero. Otherwise see from here this is a truncated exponential distribution Truncated at 0 to Peter because originally extra mental institution have support from 0 to infinity. Okay now here the likelihood function is lT to given ex girl equals two product of Aydin's from 1 to n. f. Excite Tyga. Okay So in two different minus submission X I minus. And the doctor is that Chipotle is than equals two x 1 less than infinity. I started at $3. Then it goes to extend less than infinity and targeted than zero. Okay, let's see if we first find the fighter a very bit of Of these the evacuation will be independent of theater faster at a video of this which will be independent of theater. Okay, so I will not go in that way. See we will first find for which the value of theta. The likelihood function is this maximum C do the par minus submission X. I minus in theta is maximum that implies as it is a negative sign. So this term submission extremist entities minimum. Yes minimum. Okay now this would have been minimum when she dies maximum. We'll find the maximum. Hello of peter for that. The likelihood function will be maximum and that value of theater will be the Emilio theater. So how will find the maximum value of theater? See from each equations we have three to less than equals two X. One from the first aggression. Okay from the second equation we have less than he calls two X. Two and so on. A little less than he calls to accent. See Frida is smaller than the each realization of the random sample. Okay definitely. So we'll consider here the order statistics. So let us do not. Y one records two X. Order one. That is the minimum order statistic of exciting and I won less than he calls to a less than he calls to end. Okay now see quota is smaller than they eat realization. This is definitely it will be smaller than the minimum order statistic of the random variable. Okay so this implies three days less than equals two X. Order one. Because to why 1? Because too many my mouth. Exactly. Okay so definitely so Peter is less than equals to the exalted one. That means the minimum order statistic. So we have to find the maximum value of Peter. So trita is less than that value. So that means Peter is Peter happens maximum value when it is a hello of it is the minimum model statistics. Okay so definitely we can arrive at the conclusion that had Emily Equals two. Why 1 Calls to X. Order one which is the minimum order statistic of that random one less than equals to one less than the cost. so this is the actual thank you.

Hi in this question, we are given a sample speed, see where The value of c ranges from zero to 10 and the corresponding priority. Said Function PC is integrate, oversee 1/10 days. Now, a new random variable X is created which is C square. We have to find the BDS and C D s of eggs to find the period and city about X. We know since X is because to see square C will be equal to root X. And since, see ranges from 0 to 10 C square ranges from 0 to 100. Now, in order to find the period the uhh since and another variable is created. We need to use the variable transformation formula, which is the property desert function F C G N versi into Yeah, the absolute value of D C over the eggs, plugging in all the values we will get one by 10 into one over to root x. That is one by 20 x. Now we have to find the CDF CDF is to find the syria. We will integrate the pdf over the given domain Vettetel integrate from 0 to eggs. And PDF has won over 20 route XDX. On integrating this, we will get to root X Over 20 pitches route X over 10. Hope that was helpful. Thank you

Hello. This is problem 9.83. We're going to deport a so we have a uniformly distributed uh random variable um and it's made up of wise. So firstly we need to find the likelihood, we're trying to find the Emily of data. So the likelihood of data is equal to the product of eyes. You could want to end of one over tooth data plus one. And this is the important part right here. Those brackets. So it's gonna be from zero 12 0 is less than ricotta wise, java, Which is less than or equal to two times data plus one. So this is just given us our domain at least of just one why? But now it's all of the wise. Okay so we're trying to maximize this. So um so we'll do one right? Because just 12 then it's one and then the concepts too, Times Data Plus one. We're going to exponentially ate it so too then um but now uh we need to see well when does this work? So we could just write down, okay I put a one here. And so this only works when zero is less than or equal to wise of them. She lives in our equal to two Times Data Plus one. Um So then you must be thinking what's going on here? So you want uh the numerator to be the biggest and then the biggest that he could be is if you get the maximum of the wise which is this was then um But we're not done yet so it's not wiser than the biggest. We need to do a little algebra. Uh Because what we're trying to maximize uh data. So from here we could just write down the same thing one over two times data plus 1 to the end. And then what is this called indicator function? So if you were to um subtract one and divide by two on both sides, you get negative. Uh huh. Is less than or equal to worry. So then mine is one give other way too which is less than or equal to data. So this right here would be the biggest event. So now we could just write down the Emily. So Emily of data is theater. Uh huh. Which is equal to wise event. Maya is one all over to. Okay now we need to find the Emily of the variance. So her p we know that the variants. Why Is he grew one or 12 times two data plus one square by definition and the Emily uh variants. So why is so one of the toe stays the same? Is there for me too? Times whatever data is. So instead of the theater we write down the theater. Huh Too multiplied by wise event one is one divided by two plus one. We're going to square this simplifying. We'll get wiser men squared or 12


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