Two variables have a bivariate normal distribution. Explain wh...

Question

Two variables have a bivariate normal distribution. Explain what this means.

Accepted Answer

Hello there. Today we're going to solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Which of the following statements best describes a bivariate normal distribution for two random variables A and B? Awesome. So it appears for this particular problem we're asked to look at our multiple choice answers and we're asked to determine which of our statements shown in our multiple choice answers best describes the bivariate normal distribution for two random variables A and B. So. Now that we know what we're ultimately trying to solve for, let's read off our multiple choice answers to see what our final answer might be. A is A and B are independent and both have uniform distributions. B is only A is normally distributed. C is A and B must have the same mean and variance and D is the joint distribution of A and B is normal, characterized by their means, standard deviations, and correlation. OK, so our first step in order to solve this particular problem is we need to recall a note that the bivariate normal distribution describes two variables whose joint distribution is normal. Our second step is we need to recall a note that it is defined by the means new subscript A. And new subscript B, that, so once again, it is defined by the means new subscript A and new subscript B, that the standard deviation. Sigma subscript A and sigma subscript B, and the correlation coefficient row. Between A and B. So, once again, to quickly recap, so it is defined by the means mu A and mu B, the standard deviation sigma A and sig sigma B, and the correlation coefficient row between A and B, as we We should recall. And our third step is we need to recall and note that the marginal distributions of A and B are normal, which I'll put a capital N in quotation marks to indicate that it's normal. And because the marginal distributions of A and B are normal, and any linear combination. M multiplied by A plus N multiplied by B is also normally distributed. So therefore, as a result for our fourth and final step, our correct multiple choice answer will have to be multiple choice answer letter D. Because the joint distribution of A and B is normal, characterized by their means, standard deviations, and correlation, and that is the only correct statement out of all of our multiple choice answers. And that's it. That is our final answer. So without a doubt, our final answer is multiple choice answer letter D. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Two variables have a bivariate normal distribution. Explain what this means.

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