Discrete Cumulative Distribution Functions - Extra Practice

Consider the following plot of a function \(F(x)\), which represents the Cumulative Distribution Function (CDF) of a random variable \(X\).

CDF of variable X

0.00

0.20

0.45

0.75

1.00

0

1.0

2.5

3.0

5.0

Value (x)

Which of the statements below are true? Select all that apply.

• \(P(X = 3.0) = 0.75\)
• \(P(X \le 2) = P(X = 1)\)
• \(X\) is a discrete variable that takes exactly 4 distinct values
• The probability of observing an outcome strictly greater than 3.0 is 0.25
• All values in the support of \(X\) are integer numbers

NoteShow Solution

Correct Statements:

• \(P(X \le 2) = P(X = 1)\)
• \(X\) is a discrete variable that takes exactly 4 distinct values
• The probability of observing an outcome strictly greater than 3.0 is 0.25

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Step-by-Step Breakdown

By evaluating the heights of the step discontinuities (jumps) in the graph of the Cumulative Distribution Function \(F(x)\), we can recover the individual Probability Mass Function (PMF) points:

• At \(x = 1.0\): Jump from 0.00 to 0.20 implies \(P(X = 1.0) = 0.20\)
• At \(x = 2.5\): Jump from 0.20 to 0.45 implies \(P(X = 2.5) = 0.45 - 0.20 = 0.25\)
• At \(x = 3.0\): Jump from 0.45 to 0.75 implies \(P(X = 3.0) = 0.75 - 0.45 = 0.30\)
• At \(x = 5.0\): Jump from 0.75 to 1.00 implies \(P(X = 5.0) = 1.00 - 0.75 = 0.25\)

Why statement 2 is TRUE:
Since there are no choices between \(x = 1.0\) and \(x = 2.5\), the cumulative probability anywhere inside that interval is constant. Thus, \(P(X \le 2) = F(2) = 0.20\), which perfectly matches \(P(X = 1) = 0.20\).

Why statement 4 is TRUE:
The event “strictly greater than 3.0” (\(X > 3.0\)) is the complement of \(X \le 3.0\).
\[P(X > 3.0) = 1 - F(3.0) = 1 - 0.75 = 0.25\]