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Question 3_ (4 points) random variable has. pdf given byc(1-y), if 0 < y < 1 otherwise_fly)FindFind the cumulative distribution function_Calculate ElY]-Calcul...

Question

Question 3_ (4 points) random variable has. pdf given byc(1-y), if 0 < y < 1 otherwise_fly)FindFind the cumulative distribution function_Calculate ElY]-Calculate Var[Y]Find the median of Y .

Question 3_ (4 points) random variable has. pdf given by c(1-y), if 0 < y < 1 otherwise_ fly) Find Find the cumulative distribution function_ Calculate ElY]- Calculate Var[Y] Find the median of Y .



Answers

Find (a) the mean and (b) the median of the random variable with the given pdf. $$f(x)=4 x^{3}, 0 \leq x \leq 1$$

To find the mean value of this function, we will integrate between the boundaries of 01 of X times the function three X squared d x My junk also equals, then grow from 0 to 1 of three x cubed d x but integrating. We have 3/4 x to the fourth, evaluated at 10 By applying the one and zero to the function, we simply have 3/4 times one to the fourth or one and we have 3/4 as our mean value, which also equals 0.75 in decibel. Not to find the median of the function, we will set 0.5, which is the halfway value, equal zero to some value See of the function three X squared D X. Evaluating this function it's simply results two x cubed evaluated at sea and zero find Applying the evaluation. The constant C is plugged in for X, so we have c cubed Equalling our original 0.5, raising both sides to the 1/3 power bi result in C equaling 0.7937 for the median

Find the mean off the function set mu equal to but in a girl from the boundary 01 of X f of x and we can't move the constants up front so we can get 4/1 minus. Eat the negative fourth in front of the other girl, leaving us with X. Eat the negative four x d x. You may notice this requires integration by parts, so the integration by parts formulas in a girl of you Devi equals u V minus the integral of V D u. You know, we can assign variables on the above equation. U equals X and Devi equals Eat the negative for Max dx differentiating you d u equals d eggs. Integrating TV V equals negative 1/4 e to the negative four x You know, we can play these very pulls back into the Englishman parts standard equation. Therefore, mu equals Don't forget about the constant for over one minus heeded the negative fourth. Now you ve negative X over four each of the negative for X minus the integral of VD you about your condition is 0 to 1 negative 1/4 e to the negative four ax d X. Now we can simplify and integrate. U equals constant again for over one minus eat of the negative fourth times the quantity Negative X over four e to the negative. Thank you for X now integrating. We have a positive 1/4 but we are divided by negative four which results in negative 1/16 e to the negative four X We'll evaluate between one and zero Now applying the boundary conditions, we can leave our constant outfront once again. And now just applying one. We have a negative one over for each of the negative. Four times one minus 1/16 E to the negative four times one and now applying zero to all values of X. We can't subtract. Zero since is your over 40 minus E to the zero equal one. We're left with minus 1/16. No, we can simply simplify this equation and plug in our calculator. We have the constant for over one minus eight of the fourth And since we can actually add these since they have both multiplying by E to the negative fourth. So we have negative 5/16 eat in the negative fourth, minus a negative 1/16 can add 1/16. You plug this equation into your calculator are mean equals 0.2313

Alright. For this question, we wanna find the median. Another measure of average for random variable R e to the negative are X when X is greater than or equal to zero. The media is about 50% mark on the probability that is the value M for which you were integral from M to infinity or our e to the negative R x d of X is equal to have All right now to solve this Essentially, what we do is we use that, um, average equation where we know that the integral from A to Infiniti FX G of X is the same thing as the limit as R approaches infinity for the integral from a two are here are ffx would be our r e to the negative r x. So we went ahead and wrote this out as our limit as R approaches infinity or our integral from a to our A to our we'll see that we've got I'm sorry. Not eight are should be m here m toe are or to the e negative R x dfx, we're told, is equal to one half. All right Now if you take the integral of that we'll get the limit as our approaches Infinity or e the negative R m minus e to the negative. Our capital are little are once again it is all equal to have right now if you plug in infinity as your limit we're told this are is approaching to infinity and this year will go to see Rose. You'll be left with ease equal to the negative Lower case R times and power which is equal to have Okay, now if we go ahead and try to solve for em what we're gonna do is take the l N Will have negative r m o N e is equal to our Ln of half or and then we can cancel out our l ends by making this into a negative l n two And we would have a r m negative RM here, so RM value would be the same thing as our Ln of to provided by our and then the Ellen of two was the same thing. That's 69 and that will be our final answer there. All right, well, I hope that clarifies the question there. Thank you so much for watching

First, we will find the mean from you by integrating from zero to pi over two times of f time F of X, which in this case is a sign of X d x and for this probably will have to use integration by parts The general formula for integration by parts in a girl of you Devi equals u V minus the integral V d u. So now we can assign these variables in this case you it's gonna equal. X and d V equals coast on vax dx so about differentiating you d u equals d X and integrating DV v equals sign of X. So now we can plug these variables into the Englishman parts equation. You d v something meal equals u V is x sign of X minus the neural VD you 00 to pi over two of sign of x d x and now we can simplify this equation to mu equals X sign of X minus negative co sign of access and a girl of Sinus negative co sign. We'll evaluate pi over 2 to 0. Evaluating and simple find mu well equal putting in pi over to first pi over two and the sign of pi over two equals one. Adding the coastline a pyre were too, which equals zero Now, playing in zero zero times zero times signing 00 plus co. Sign of zero equals one. So now we can simplify a little bit further Is mu equals pi over two minus one. Therefore, mu equals negative pi over two for the mean not to solve. For the median, it's 0.5 equal to the integral from zero to some constant C of the original function F of X, which is co sign of X dx. Evaluating this integral 0.5 equals that anger of coastline, which is sign of X and simply we can evaluate at sea and zero. And by simplifying this equation, 0.5 can end, see, and for all the accents. So it was sinoussi minus sign of zero, and science zero is equal to zero. So we have that 0.5 equals sign of C. So two sulphur see, we will apply the inverse sign to both signs in verse. Sign of 0.5 equals the inverse sign of sign of C, which is just see and Denver. Sign of 1/2 is equal to pi over six for our median. That is the final answer


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