请老师解释一下B选项

问题如下：

Question

Which of the following statements is correct?

A company forecasts that net income next year will range from a loss of $4,000,000 to a gain of $12,000,000. The graph shows the probability density function for this continuous distribution.

选项：

A.The probability of avoiding a loss is 0.25. B.The expected net income is $8,000,000. C.The probability of incurring a loss of $2,000,000 is 0.

解释：

Solution

C is correct. For a continuous uniform random variable, all outcomes are equally likely and the probability of the random variable assuming any given fixed value, such as $2,000,000, is zero.

A is incorrect because the probability of avoiding a loss is the area under the probability density function from zero to $12,000,000 = P(0 ≤ X ≤ 12). It includes an area with a base of 12 and a height of 1/16. Base times height = 12/16 = 75.0% of the total rectangular distribution function.

B is incorrect because, as a continuous uniform random variable, the expected value (mean) for net income is the midpoint between the lowest and highest values:

μ = (a + b)/2

= (-$4,000,000 + $12,000,000)/2

= $8,000,000/2

= $4,000,000