Hi, I don\'t get how to get the highlighted part. Specifically, don\'t know hpw

Question

Hi, I don't get how to get the highlighted part. Specifically, don't know hpw E(ZX) - E(Z)E(X) became that intagral - can someone kindly show intermediate steps using the distribution/whatever of Z and X?
5· Let Z be a standard normal random variable. Define X to be Bernoulli such that x=1 if Z>c where c is a given constant. Compute the correlation between Z and X. Simplify your answer as much as possible. Solution me 2 where (x) is a normal density function Corr(Z, X) = , Cov(ZA) Note that conditioning on Z will not lead to a shorter solution. For example, (c) 2dx = wl

Explanation / Answer

Hello,

So basically here Z is a standard normal variable and X is a Bernoulli.

Now hopefully you know that the standard normal distribution has mean 0 and variance 1. Therefore E[Z]=0 according to the basic norms. Therefore the term E[Z]*E[X]=0.

Hence, the expectation of the joint distribution, E[XZ] will be the product of the bernoulli, i.e, x multiplied by the standard normal distribuition hence you get that line which you have underlined.

In the next part that you have asked, Correlation Function is a part of Time Series, ie, in a subpart of Autocorrelation.

Now from the laws of Binomial Distribution, we know that Variance = n*p*(1-p) which we have using since ages. You can actually visualise this from the same logic how we visualized in the case of Binomial and Normal Distribution.

Hope it helped. Cheers!

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