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Problem 119 Hard Difficulty

Let $X$ and $Y$ be the times for a randomly selected individual to complete two different tasks, and
assume that $(X, Y)$ has a bivariate normal distribution with $\mu_{1}=100, \sigma_{1}=50, \mu_{2}=25,$$\sigma_{2}=5, \rho=.4 .$ From statistical software we obtain $P(X<100, Y<25)=.3333, P(X<50,$ $Y<20 )=.0625, P(X<50, Y<25)=.1274,$ and $P(X<100, Y<20)=.1274 .$
(a) Determine $P(50<X<100,20<Y<25)$ .
(b) Leave the other parameters the same but change the correlation to $\rho=0$ (independence).
Now recompute the probability in part (a). Intuitively, why should the original be larger?

Answer

a. p(50 < x < 100 , 20 < y< 25) = .1410 b. 1165

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Video Transcript

Hi, guys. I'm gonna explain the solution off number 119 But let me explain a little bit about probability off normal distribution on some basic laws and general probability. So if you wanna come, come compute their probability off. For example, here ability of X MSM 50. We can trust their toe what we call standard normal distribution. I have probability off X minus mu divided by standard deviation Onda. We can find these values either from calculator or standard a normal distribution tables on the second thing that I want a mention here is we have two basic formulas for ah, the probability off A A and B given the situation that we have. So, in case of independent events, um, the privilege is gonna be ah, basically probability of a multiplied by B. Andi. Think that's gonna be in part be on our question on, in case off we have some some inclusive events is gonna be provocative. A plus probability of B minus, um, the the Union Probability. And here's an explanation off Vin diagram. What does ah, on mean? And I'm gonna come up with, uh, with a solution for that for the normal solution here, but in a minute. So for part A, we are trying to find the probability off excess between 1500 on why is between 20 and 25 on? We know that it has some correlations. So we are in the second case here for a so simply you have to fold these formula. Okay, so that's haven't a B and Houston Union, and you'll find that that is the final social for part B. We want toe have the correlation is equal to zero, which means independence. So let me explain that a normal distribution here, So if we have a value, is exactly on the mean Okay, it's gonna be 50% on with this one. Standard deviation below the mean is gonna be 34%. Given that, um, we've been asked toe find these guys on Heaven's one of them is on the mean and the other guy is, uh, on does the same here. That's the mean and one standard deviation below that. So it is gonna be the probability born three for 13 I just quality for simplicity on is gonna be team out of liberty because we haven't independent events so it's gonna be a T Square. And here's the final answer. And as you notice this quantity is, let's done whatever we got from, uh, uh, Park be because of the ah, the correlation is less and embark. Be coincide with the law.