If events x and y are independent, what must be true? check all that apply. p(y | x) = 0 p(x | y) = 0 p(y | x) = p(y) p(y | x) = p(x) p(x | y) = p(y) p(x | y) = p(x)

3 and 6 are correct

Step-by-step explanation:

if events X and Y are independent, then for intersection we multiply the probability

P(Y∩X)  = P(Y) * P(X)

We know that

Now we replace P(Y) * P(X) for  P(Y∩X)

Cancel out P(X)

So

Like that

Now we replace P(X) * P(Y) for  P(X∩Y)

Cancel out P(Y)

So

P(Y | X) = P(Y)  and P(X | Y) = P(X)  are true

option a.

step-by-step explanation:

if you look closely at the graph, we can notice that the function is not defined at two different points.

if you look at functions b and c, you will notice that both functions are not defined at just one point. the function b is not defined at x= -5/2 and function c is not defined at x=-5. both options are discarded, given that our function should not be defined at two fiferent points.

now, if you look closely at the graph you will notice that the function sketched is not defined at x=5 and x=-2.

pakistan's principal natural resources are arable land and water. about 25% of pakistan's agriculture accounts for about 21% [1] of gdp and employs about 43% of the labour force. in pakistan, the most agricultural province is punjab where wheat and cotton are the most grown.