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# Correlation of two ab(Normals) | ISI MStat 2016 PSB Problem 6 This problem is an interesting application of the moment generating function of normal random variable to see how the correlation behaves under monotone function. This is the problem 6 from ISI MStat 2016 PSB.

## Problem

Suppose that random variables and jointly have a bivariate normal distribution with and
correlation . Compute the correlation between and .

### Prerequisites

• Correlation Coefficient
• Moment Generating Function
• Moment Generating Function of Normal ~ is .

## Solution is called the moment generating function.

Now, let's try to calculate For, that we need to have the following in our arsenal.

• ~ • ~ • ~ [ We will calculate this just below ]. .

Now observe the following:

• • • • #### Important Observation and always have the same sign. Can you guess why? There is, in fact, a general result, which we will mention soon.

Now, we are left to calculate . .

Therefore, .

Observe that the mininum correlation of and is .

## Back to the important observation and always have the same sign. Why is this true?

Because, is an increasing function. So, if and are positively correlated then, as increases, also increases in general, hence, also increases along with hence, the result, which is quite intuitive.

Observe that in place of if we would have taken, any increasing function , this will be the case. Can you prove it?

Research Problem of the day ( Is the following true? )

Let be an increasing function of , then

• • • This problem is an interesting application of the moment generating function of normal random variable to see how the correlation behaves under monotone function. This is the problem 6 from ISI MStat 2016 PSB.

## Problem

Suppose that random variables and jointly have a bivariate normal distribution with and
correlation . Compute the correlation between and .

### Prerequisites

• Correlation Coefficient
• Moment Generating Function
• Moment Generating Function of Normal ~ is .

## Solution is called the moment generating function.

Now, let's try to calculate For, that we need to have the following in our arsenal.

• ~ • ~ • ~ [ We will calculate this just below ]. .

Now observe the following:

• • • • #### Important Observation and always have the same sign. Can you guess why? There is, in fact, a general result, which we will mention soon.

Now, we are left to calculate . .

Therefore, .

Observe that the mininum correlation of and is .

## Back to the important observation and always have the same sign. Why is this true?

Because, is an increasing function. So, if and are positively correlated then, as increases, also increases in general, hence, also increases along with hence, the result, which is quite intuitive.

Observe that in place of if we would have taken, any increasing function , this will be the case. Can you prove it?

Research Problem of the day ( Is the following true? )

Let be an increasing function of , then

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