For a discrete random variable, a student observes the CDF and interprets F(3) as 'the probability that X equals exactly 3.' What does F(3) actually represent?
AThe probability that X equals 3, equivalent to the PMF value f(3)
BThe total accumulated probability that X is 3 or less: P(X ≤ 3)
CThe probability density at x = 3 (valid for continuous distributions)
DThe probability that X is greater than 3
F(3) = P(X ≤ 3), the sum of all probability mass at or below 3. The PMF f(3) gives only P(X = 3). The CDF is a running total — the jump at x = 3 equals f(3), but F(3) itself includes all probability mass at x ≤ 3. Conflating the CDF with the PMF is the most common error in working with distributions.
Question 2 Multiple Choice
The CDF of a continuous random variable is F(x) = x² for 0 ≤ x ≤ 1. What is the probability density function f(x) on this interval?
Af(x) = x² (the CDF and PDF are the same function)
Bf(x) = 2x (the derivative of the CDF)
Cf(x) = √x (the square root of the CDF)
Df(x) = 1/(2x) (the reciprocal of the derivative)
For a continuous random variable, f(x) = F'(x). Differentiating x² gives 2x. This relationship — the PDF is the derivative of the CDF, or equivalently the CDF is the antiderivative of the PDF — is the key connection between the two representations and means you can convert in either direction.
Question 3 True / False
The CDF can decrease as x increases, since probability accumulated earlier can be 'redistributed' to later regions.
TTrue
FFalse
Answer: False
The CDF F(x) = P(X ≤ x) is always non-decreasing. As x increases, more values fall at or below x, so the accumulated probability can only stay the same or increase. A decreasing CDF would imply negative probability, which is impossible. The CDF must go from 0 at −∞ to 1 at +∞ monotonically.
Question 4 True / False
For any random variable — whether discrete or continuous — the probability P(a < X ≤ b) can be computed as F(b) − F(a).
TTrue
FFalse
Answer: True
This is one of the most useful properties of the CDF. P(a < X ≤ b) = P(X ≤ b) − P(X ≤ a) = F(b) − F(a). This works for both discrete and continuous distributions. For discrete variables, care is needed with strict vs. non-strict inequalities at isolated points, but the formula P(a < X ≤ b) = F(b) − F(a) holds exactly as written.
Question 5 Short Answer
Explain why the CDF of a discrete random variable forms a staircase shape, and what determines the height of each jump.
Think about your answer, then reveal below.
Model answer: The CDF is flat between support values because no new probability accumulates there, and jumps at each support value by exactly P(X = x) — the PMF value at that point.
Between the discrete values where X can take on mass, F(x) = P(X ≤ x) does not change since no additional probability is accumulated. At each support point x₀, the CDF jumps up by f(x₀) = P(X = x₀), which is the probability mass at that value. This is why the jump sizes encode the PMF, and summing all jumps gives a total of 1.