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Option 2 is correct.

Binomial Distribution approaches Gaussian distribution for large n. And the Gaussian distribution with mean np and variance npq.

Basically, these are the mean and variance of the Binomial Distribution.

Now if X ~ Gaussian( np, npq )

Then, the question asks what Random Variable in terms of X will have Gaussian ( 0, 1). Because Normal Distribution is nothing but normalized Gaussian distribution.

So, if we subtract mean from X, then :

X - np ~ Gaussian ( 0, npq )

And, then divide by standard deviation to achieve varince 1 :

( X-np)/√npq ~ Gaussian ( 0, 1) , which is Normal Distribution.

So, correct option is option 2
by (339 points)

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