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#### marcadams267

##### New member

- Aug 26, 2019

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Consider two Bernoulli processes X1 and X2 such that

X1[k] is a Bernoulli random variable with P=0.5 and

X2[k] is a Bernoulli random variable with P=0.7 for all k>=0

Let Y be a random process formed by merging X1 and X2, i.e. Y[k] =1 if and only if X1[k] = X2[k] = 1 and Y[k] = 0 otherwise.

a.) Solve for the success probability of Y if X1 and X2 are uncorrelated.

b.) Solve for the success probability of Y if E[X1[k]X2[k]] = 0.3.

c.) If E[X1[k]X2[k]] is constant for all k, find the minimum possible success probability of Y.

d.) If E[X1[k]X2[k]] is constant for all k, find the maximum possible success probability of Y.

My initial thought was that the success probability of Y was just the intersection of x1 and x2 (when both are equal to 1) or 0.7*0.5 = 0.35.

However, the way to problem is written suggests that correlation affects the probability. Can someone explain how this is so?

Edit: I've realized that the 2 bernoulli processes may not be independent, therefore, my initial understanding of the problem is wrong.

However, how do I use the information given (the correlation) to solve for the probability? What equation relates the two together, or what assumptions can I make about the problem given the correlation?

X1[k] is a Bernoulli random variable with P=0.5 and

X2[k] is a Bernoulli random variable with P=0.7 for all k>=0

Let Y be a random process formed by merging X1 and X2, i.e. Y[k] =1 if and only if X1[k] = X2[k] = 1 and Y[k] = 0 otherwise.

a.) Solve for the success probability of Y if X1 and X2 are uncorrelated.

b.) Solve for the success probability of Y if E[X1[k]X2[k]] = 0.3.

c.) If E[X1[k]X2[k]] is constant for all k, find the minimum possible success probability of Y.

d.) If E[X1[k]X2[k]] is constant for all k, find the maximum possible success probability of Y.

My initial thought was that the success probability of Y was just the intersection of x1 and x2 (when both are equal to 1) or 0.7*0.5 = 0.35.

However, the way to problem is written suggests that correlation affects the probability. Can someone explain how this is so?

Edit: I've realized that the 2 bernoulli processes may not be independent, therefore, my initial understanding of the problem is wrong.

However, how do I use the information given (the correlation) to solve for the probability? What equation relates the two together, or what assumptions can I make about the problem given the correlation?

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