Question

Suppose x is uniformly distributed on {a, a 1, a 2, ..., b}. If E[x] = 2.5 and Var[x] = 1.25, what is Pr[x>2]?

a) 0.25
b) 0.50
c) 0.75
d) 1.00

Answer 1

To find the probability that x > 2 in a uniform distribution, we calculate the area under the probability density function (PDF) from 2 to the upper limit, which is b. The probability is equal to the height of the PDF (1/(b - a)) multiplied by the width (b - a). In this case, the probability is 2/3.

Step-by-step explanation:

To find the probability that x>2, we need to calculate the area under the probability density function (PDF) from 2 to the upper limit, which is b. Since x is uniformly distributed, the PDF is a rectangle with height 1/(b - a) and width (b - a). In this case, the width is 1, as the question states that x is distributed on {a, a1, a2, ..., b}. The mean (E[x]) of a uniform distribution is given by (a + b)/2, which is 2.5. The variance (Var[x]) is [(b - a)^2]/12, which is 1.25. Solving for a and b, we get a = 1 and b = 4.

Now, let's calculate the probability that x > 2. The probability is equal to the area under the rectangle from 2 to 4. Since the PDF of a uniform distribution is constant, the area of a rectangle is equal to its height times its width. In this case, the height is 1/(b - a) = 1/3. So, the probability is (1/3) * (4 - 2) = 2/3.

Therefore, the answer is not one of the given options. The correct probability is 2/3.