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(6 pts) continuous random variable X has depsity (pdf)f(z) 9 22 0 < = < 3. Find the median value of X_ This is defined to be the number m satisfying the equat...

Question

(6 pts) continuous random variable X has depsity (pdf)f(z) 9 22 0 < = < 3. Find the median value of X_ This is defined to be the number m satisfying the equation F(m) = P[X < m] = 0.5, where F is the cdf of X_10. (6 pts) Suppose X is a random variable whose moment generating function is 25e2t mx (t) 35et)2Find E[X]-

(6 pts) continuous random variable X has depsity (pdf) f(z) 9 22 0 < = < 3. Find the median value of X_ This is defined to be the number m satisfying the equation F(m) = P[X < m] = 0.5, where F is the cdf of X_ 10. (6 pts) Suppose X is a random variable whose moment generating function is 25e2t mx (t) 35et)2 Find E[X]-



Answers

Identify the probability density function. Then find the mean, variance, and standard deviation without integrating. $$ f(x)=\frac{1}{6 \sqrt{2 \pi}} e^{-(x-30)^{2} / 72},(-\infty, \infty) $$

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

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

From were given a pdf over this interval, and we need to find the mean, which is the expected value of a random variable X and the median. So first finding the mean Well, what's the definition of expected value? It's our pdf Times X. We integrate that over the region. So we have excused here, and this is giving us a kn anti derivative x of the fourth over 36 evaluated between three and zero, which is giving us three of the fourth over I can write that is three squared times four. So that should reduce to 9/4. That's the expected value. And now what is the definition of the median? Well, it's the point in this interval where the probability that she random variable was less than the median is 1/2 and vice versa. Probably that's greater than meeting is 1/2. So to find the median, let's set up this equation. 1/2 is equal to the integral from zero to some point M. That's the median that we're looking for of 1/9 of her X squared. So if we find the anti derivative here we get X cubed over 27 evaluated between em and zero gives us m cubed over 27. And so now the median should be between zero and three. So let's take the positive cube root of 27 over too, which on approximate value for that is 2.38

Yeah. So the expectation help x squared echoes the country girl from 0 to 8 ax squared times the probability density function Which recalls stray over 256 Integral from 0 to 8 eight X. Cubed minus X. Force dance. Yes it goes way over 256 two X 2/4 minus extra face over five. Hear from 0- eight. And this equals 19 points to. And the expectation of X cubed Because integral from 0 to AIDS. That's cubed times probability density function. And they're seacoast Stray over 256 Integral from 0 to 8 eight X. to force miners extra face. And this cycle stray over 256 times 8/5 X 2/5 minus X 2 6/6. And from 0 to 8. And the answer areas Hand Guardian too. 0.4.


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