Q1. Which of the following qualifies as a cumulative distribution function?

A)   F(1) = 0, F(2) = 0.5, F(3) = 0.5, F(4) = 0.

B)   F(1) = 0.5, F(2) = 0.25, F(3) = 0.25.

C)   F(1) = 0, F(2) = 0.25, F(3) = 0.50, F(4) = 1.

Q2. A random variable X is continuous and bounded between zero and five, X0 ≤ X ≤ 5). The cumulative distribution function (cdf) for X is F(x) = x / 5. Calculate P(2 ≤ X ≤ 4).

A)  0.40.

B)  1.00.

C)  0.50.

答案和详解如下：

Q1. Which of the following qualifies as a cumulative distribution function?

A)   F(1) = 0, F(2) = 0.5, F(3) = 0.5, F(4) = 0.

B)   F(1) = 0.5, F(2) = 0.25, F(3) = 0.25.

C)   F(1) = 0, F(2) = 0.25, F(3) = 0.50, F(4) = 1.

Correct answer is C)

Because a cumulative probability function defines the probability that a random variable takes a value equal to or less than a given number, for successively larger numbers, the cumulative probability values must stay the same or increase.

Q2. A random variable X is continuous and bounded between zero and five, X0 ≤ X ≤ 5). The cumulative distribution function (cdf) for X is F(x) = x / 5. Calculate P(2 ≤ X ≤ 4).

A)  0.40.

B)  1.00.

C)  0.50.

Correct answer is A)

For a continuous distribution, P(a ≤ X ≤b) = F(b) − F(a). Here, F(4) = 0.8 and F(2) = 0.4. Note also that this is a uniform distribution over 0 ≤ x ≤ 5 so Prob(2 < x < 4) = (4 − 2) / 5 = 40%.

cc

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