Question

1. Suppose that 2000 points are selected independently and at random from the unit square, {(x, y)|0 ? x ? 1, 0 ? y ? 1}. Let the random variable W be the number of such points that fall in the set A = {(x, y)|x² + y² < 1, x ? 0, y ? 0} a. For a given point, what is the probability that the point will be inside the set A? Hint: You need to use some geometry here (2 points) b. How is W distributed (that is, give the name of the distribution and the parameters)? Give the PMF of W (3 points) c. Give the mean, variance, and standard deviation of W (3 points) d. Find the probability that 1250 of the randomly selected points are in A (2 points)

          1. Suppose that 2000 points are selected independently and at random from the unit square, {(x, y)|0 ? x ? 1, 0 ? y ? 1}. Let the random variable W be the number of such points that fall in the set A = {(x, y)|x² + y² < 1, x ? 0, y ? 0}
a. For a given point, what is the probability that the point will be inside the set A? Hint: You need to use some geometry here (2 points)
b. How is W distributed (that is, give the name of the distribution and the parameters)? Give the PMF of W (3 points)
c. Give the mean, variance, and standard deviation of W (3 points)
d. Find the probability that 1250 of the randomly selected points are in A (2 points)

1. Suppose that 2000 points are selected independently and at random from the unit square, (x, y)|0 ? x ? 1, 0 ? y ? 1. Let the random variable W be the number of such points that fall in the set A = (x, y)|x² + y² < 1, x ? 0, y ? 0
a. For a given point, what is the probability that the point will be inside the set A? Hint: You need to use some geometry here (2 points)
b. How is W distributed (that is, give the name of the distribution and the parameters)? Give the PMF of W (3 points)
c. Give the mean, variance, and standard deviation of W (3 points)
d. Find the probability that 1250 of the randomly selected points are in A (2 points)

Added by Manuela J.

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