Question

If Z is a bernoulli random variable such that point (X,Y) falls within unit circle with center (0,0) and given that X & Y are independent uniform random variables with range [-1,1] and X^2 + Y^2 <1. Find the expectation E(Z) and Std. Deviation of Z.

Answer 1

A Bernoulli random variable has only two outcomes either success or failure. Let P(0) indicate the probability that the point doesn't fall within the unit circle and P(1) indicate the probability that the point falls within the unit circle.

Given X and Y are an independent uniform random variables with a range [-1, 1].

Therefore (X, Y) can be any point in a square with center (0, 0) and the side length of 2 units.

P(1) = Probability that point lies within unit circle = Area of unit circle / Area of square = = /4

P(0) = Probability that point doesn't lie within unit circle = 1 - P(0) = 1 - /4

Let p be the probability of success(Here P(1))

Then the expectation of bernoulli random variable is p and standard deviation of bernoulli random varibale is

Expectation = p = P(1) = /4 = 0.785

Standard deviation = = = 0.411