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5) A distribution has a density function f(x) = X'* 2 1 a) Determine the ML (Maximum Likelyhood) of parameters @ based on the observations from the distributio...

Question

5) A distribution has a density function f(x) = X'* 2 1 a) Determine the ML (Maximum Likelyhood) of parameters @ based on the observations from the distribution:b) Calculate P(X < 2) given the observations x1 = 5, xz = 1, Xa = 4, X4 = 3

5) A distribution has a density function f(x) = X'* 2 1 a) Determine the ML (Maximum Likelyhood) of parameters @ based on the observations from the distribution: b) Calculate P(X < 2) given the observations x1 = 5, xz = 1, Xa = 4, X4 = 3



Answers

Let $X$ be a continuous random variable with values between $A=1$ and $B=\infty,$ and with the density function $f(x)=4 x^{-5}.$
(a) Verify that $f(x)$ is a probability density function for $x \geq 1.$
(b) Find the corresponding cumulative distribution function $F(x).$
(c) Use $F(x)$ to compute $\operatorname{Pr}(1 \leq X \leq 2)$ and $\operatorname{Pr}(2 \leq X).$

Yeah that's probably have been given the cumulative distribution function. Capital F of X is equal 20. If X is less than zero. 0.2 Acts effects is between zero and five and one. Mhm. If X is greater than or equal to five and we would like to find a few different probabilities from us. Yeah, I would first like to find the probability That X is less than 2.8. Well this is just the definition of capital f. 2.8 mm Which is 0.2 times 2.8. We just plug it infrared. What's going on here? All we're going to do is take .2 times 2.8 and that's going to give us this probability and when we do This gives us .56. So these probabilities .56? Yeah, on b We would like to find the probability X is greater than 1.5. Well this is equal to 1- the probability X is less than or equal to 1.5 Which is 1 -101.5 Which is 1 -0.2 times 1.5. Mhm. It's always that 1 -2 times 1.5. And this gives us .7. On c we want the probability X is less than -2. Again this is just capital f of -2. And from the chart since this is less than zero this is just zero. Mhm. And um do you want the probability X is greater than six. Yeah Which is 1- the probability that X is less than or equal to search Just one -F of six. If such as ones, this is 1 -1 which is zero.

Hi. So I'm gonna go over, um, answering a question about probability density function using some calculus. So we are given a probability density function. P of X is equal to 1/4 times of one plus X. That quantity raised to the negative five horsepower. And we're also being told that the domain for that function is from zero to infinity. The questions asking us to figure out what the probability is of having some X value between 0 to 2. Secondly, what's the probability of randomly getting some X value between 123? And finally, what's what's the probability of some random X value being five for bigger, greater than 45? But people, right, answer those three questions. We do want to test the validity of this density function. First, there's a couple of conditions that must be met. First of all, we have to be sure that when we integrate p of X, we get non negative values. And from this domain that is given from zero to infinity, that condition is met. The second condition is when we integrate this density probability function over the entire domain. We have to come up with one which means there's 100% chance that the X value that's gonna end up being chosen has to be in the domain of this function. So let's go ahead and look at that second validation. So I'll take the limit as be approaches infinity, since this is an improper integral of zero to be a p of x, t x and p of X was given to us as one port of one plus X raised to the negative five force and well in Great that with respect to X, I'll go ahead and continue looking at the limit as be approaches infinity. The integration of p of X would be 14 of one plus X, add one to the exponents and then divide everything by the new exponents. Dividing by negative 1/4 is simply the same as multiplying by negative for, and I'll go ahead and clean that up a little bit. So we're looking, um, at the limit as he approaches infinity of negative one over the square root of one plus X. And that would be not the square root, but the fourth route, and we're going to evaluate that over the entire domain given for this function, which would be from zero to be okay, so come up here and again. We'll keep using the limit until there's no more bees. In my expression, I'm gonna have negative one over the fourth root of one plus B minus negative one overthe e fourth root of one plus zero is one. And what we know is as B goes towards infinity, this denominator will go to infinity and the limit of negative one over infinity. We'll head towards zero. So we get zero minus negative. One divided by the fourth fruit of one is negative one and of course, zero minus negative. Wine gives us one or what would be considered 100%. So our second test is met. This is a valid density function of pot of probability so we can go ahead and now answer these questions A B and C case apart again is asking us to figure out what's the probability of randomly getting an X value between 0 to 2 inclusive and on the right here, I've just kept in mind what the indefinite integral of p of X t X is just making it easier on myself. So what we're going to do now is we're gonna add up all the probabilities or integrate not over the entire domain, but just the subset from 0 to 2, which the hard part's already done for us. I already know the indefinite integral of P of X T X is one over the fourth root of one plus x plus C. But I don't need the plus C because now it is a definite integral going from 0 to 2. So we just have to substitute and evaluate using fundamental therm of calculus. I'm gonna have negative one over the fourth root of one plus two is three minus negative, one all over the fourth root of one. Plus, there's one in the fourth grade of one is one. But put that my calculator and evaluate it really carefully. I'm going to get about 0.240164 which tells me that the probability of randomly ending up with an expel you from 0 to 2 inclusive is just about 24.16%. Okay, let's move on to the second part of the question, which is asking us what's the probability of I'm trying to see the X value will end up being from any real number. 123 Okay, so I'll bet you can guess what's gonna happen here. We need to integrate all the probabilities, Adam all up for all real numbers. But instead of going from 0 to 2 are limits of integration are going to go from 1 to 3. Well, that's not bad, because I already know that the integration is gonna be one divided by sorry, negative one divided by the fourth root of one plus X. But now the definite integral is going from 1 to 3. So this just simply means negative one, divided by the fourth root of four minus negative, one divided by the fourth root of two. And again, just be really careful when you're putting these values in your calculator. This should come up to be approximately zero point 133 seven, 89 six. And so we could say the probability that X is going to end up being any real number from 1 to 3. Inclusive is about 13 point three seven 9% and typically we go out to the nearest 1000. All right, I think we're ready for part C. That's the probability of getting any number five or greater


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