How Can We Minimize Prediction Errors in Binary Variable Analysis?

[Luksdoc]

Homework Statement

Let Y be binary variable P(Y = 1) = P(Y = 0) = 0.5 and X a random variable uniform on [0,5] when Y = 0 and uniform [4, 9] when Y = 1. Draw mean of X and P(Y = 1|X = x) as functions of x. What is the minimum probability of rejection to predict Y from X without mistake.

Homework Equations

The Attempt at a Solution

Since Y acts like some "switch", I considered two independant distributions of X given Y: p = 1/5 on [0,5] (for Y = 0) and the other one p = 1/5 on [4,9] (for Y = 1). So two means are: 2.5 (for Y = 0) and 7.5 for (for Y = 1).

For P(Y = 1|X = x):
if X in [0, 4]: P(Y = 1|X = x) = 0
if X in [5, 9]: P(Y = 1|X = x) = 1
if X in [4, 5]: P(Y = 1|X = x) = 0.5

for this "What is the minimum probability of rejection to predict Y from X without mistake." I have no idea.

 

[vela]

Homework Statement

Let Y be binary variable P(Y = 1) = P(Y = 0) = 0.5 and X a random variable uniform on [0,5] when Y = 0 and uniform [4, 9] when Y = 1. Draw mean of X and P(Y = 1|X = x) as functions of x. What is the minimum probability of rejection to predict Y from X without mistake.

Homework Equations

The Attempt at a Solution

Since Y acts like some "switch", I considered two independent distributions of X given Y: p = 1/5 on [0,5] (for Y = 0) and the other one p = 1/5 on [4,9] (for Y = 1). So two means are: 2.5 (for Y = 0) and 7.5 for (for Y = 1).

I think the problem is asking you for E[X] as a function of x, similar to what you did for P(Y=1|X=x). If x is in [0,4], what is E[X]? And so on.

 

[Ray Vickson]

Homework Statement

Let Y be binary variable P(Y = 1) = P(Y = 0) = 0.5 and X a random variable uniform on [0,5] when Y = 0 and uniform [4, 9] when Y = 1. Draw mean of X and P(Y = 1|X = x) as functions of x. What is the minimum probability of rejection to predict Y from X without mistake.

Homework Equations

The Attempt at a Solution

Since Y acts like some "switch", I considered two independant distributions of X given Y: p = 1/5 on [0,5] (for Y = 0) and the other one p = 1/5 on [4,9] (for Y = 1). So two means are: 2.5 (for Y = 0) and 7.5 for (for Y = 1).

For P(Y = 1|X = x):
if X in [0, 4]: P(Y = 1|X = x) = 0
if X in [5, 9]: P(Y = 1|X = x) = 1
if X in [4, 5]: P(Y = 1|X = x) = 0.5

for this "What is the minimum probability of rejection to predict Y from X without mistake." I have no idea.

I don't see any way of predicting Y exactly from X in all cases. If we happen to observe a value of X between 4 and 5, Y is allowed to be 0 or 1, and there is no way to be sure which is correct.

RGV