Understanding Cumulative Distribution Functions Quiz

1) A student claims “Since F(2) = 0.7, the probability that X equals 2 is 0.7.” What is the best correction?

Correct, because CDF gives the probability at 2

Incorrect; probabilities cannot exceed 0.5

Correct if X is continuous

Incorrect; the PMF/PDF gives probability at a point, while the CDF gives P(X ≤ 2)

CDF gives total probability up to 2, not exactly at 2

Explanation

CDF gives total probability up to 2, not exactly at 2

2) If F is a valid CDF, which inequality must always hold?

F(x) ≥ 1 for large x

F(x) ≤ 0 for small x

0 ≤ F(x) ≤ 1 for all x

F(x) alternates above and below 0

CDF values always stay between 0 and 1

Explanation

CDF values always stay between 0 and 1

3) Which scenario most directly uses a CDF to make a decision (HSS.MD.A.3)?

Choosing the mean of data to summarize a sample

Drawing a histogram to visualize spread

Computing variance to compare two samples

Using F(t) to find P(X ≤ t) to decide a cutoff for passing a test

CDFs help find probabilities like cutoffs or thresholds for decisions

Explanation

CDFs help find probabilities like cutoffs or thresholds for decisions

4) What is F(0)?

0.10

0

0.35

1.00

F(0) = P(X ≤ 0) = 0.10

Explanation

F(0) = P(X ≤ 0) = 0.10

5) What is F(1)?

0.10

0.35

0.25

0.45

F(1) = P(X ≤ 1) = 0.10 + 0.25 = 0.35

Explanation

F(1) = P(X ≤ 1) = 0.10 + 0.25 = 0.35

6) What is F(2)?

0.40

1.00

0.75

0.65

F(2) = P(X ≤ 2) = 0.10 + 0.25 + 0.40 = 0.75

Explanation

F(2) = P(X ≤ 2) = 0.10 + 0.25 + 0.40 = 0.75

7) What is F(3)?

0.90

0.75

0.25

1.00

F(3) = 0.10 + 0.25 + 0.40 + 0.25 = 1.00

Explanation

F(3) = 0.10 + 0.25 + 0.40 + 0.25 = 1.00

8) Which of the following equals P(1 < X ≤ 3)?

F(3) − F(1)

F(2) − F(0)

F(3) − F(2)

F(1) − F(3)

P(1

Explanation

P(1

9) Which statement is true about the CDF F(x) for this X?

F(x) can decrease when x increases

F(x) stays constant except at x-values with positive probability

F(x) is the same as the PMF

F(x) must jump to values bigger than 1 if x is large

CDF only increases where there is probability, otherwise stays constant

Explanation

CDF only increases where there is probability, otherwise stays constant

10) Which best defines a cumulative distribution function (CDF)?

The function that gives the exact probability at each single value x

The function F(x) = P(X = x)

The function F(x) = P(X ≤ x)

The function F(x) = P(X ≥ x)

A CDF gives the probability that X is less than or equal to x

Explanation

A CDF gives the probability that X is less than or equal to x

11) For any random variable X, which of the following is always true about its CDF F(x)?

F(x) can take negative values

F(x) is periodic in x

F(x) sums to 1 across all x

F(x) is nondecreasing in x

CDF never decreases as x increases

Explanation

CDF never decreases as x increases

12) Which statement distinguishes a PDF/PMF from a CDF?

A CDF accumulates probability up to x; a PMF/PDF gives probability at x

A PMF/PDF accumulates probability up to x; a CDF gives probability at x

Both a CDF and PMF/PDF give the same value

Only the CDF can be used to find probabilities

CDF adds up probabilities up to x, PMF/PDF gives the value at x

Explanation

CDF adds up probabilities up to x, PMF/PDF gives the value at x

13) For a continuous random variable with density f(x), which relation is correct?

F(x) = f(x)

F′(x) = f(x) where differentiable

F′(x) = −f(x)

F(x) = ∫x∞ f(t) dt

The derivative of the CDF equals the PDF

Explanation

The derivative of the CDF equals the PDF

14) If F is a CDF, which values must it approach as x → −∞ and x → ∞?

1 as x → −∞ and 0 as x → ∞

0 as x → −∞ and 0 as x → ∞

0 as x → −∞ and 1 as x → ∞

1 as x → −∞ and 1 as x → ∞

CDF starts at 0 and approaches 1 as x increases

Explanation

CDF starts at 0 and approaches 1 as x increases

15) Which equation correctly expresses P(a < X ≤ b) in terms of the CDF F?

F(a) − F(b)

F(b) + F(a)

1 − F(b) + F(a)

F(b) − F(a)

P(a

Explanation

P(a

16) What is P(Y ≤ 1)?

0.2

0.4

0.5

1.0

F(1) = 0.4 × 1 = 0.4, but since 0 ≤ y ≤ 2.5, it’s 0.4×(1/2) = 0.2

Explanation

F(1) = 0.4 × 1 = 0.4, but since 0 ≤ y ≤ 2.5, it’s 0.4×(1/2) = 0.2

17) What is P(1 < Y ≤ 2)?

0.2

0.4

0.6

0.8

F(2)−F(1) = (0.4×2) − (0.4×1) = 0.8−0.4 = 0.4 (≈0.2 range probability per unit)

Explanation

F(2)−F(1) = (0.4×2) − (0.4×1) = 0.8−0.4 = 0.4 (≈0.2 range probability per unit)

18) What is P(Y > 2.5)?

0

0.2

0.5

0.8

For y > 2.5, F(y) = 1, so P(Y > 2.5) = 1−1 = 0

Explanation

For y > 2.5, F(y) = 1, so P(Y > 2.5) = 1−1 = 0

19) Which statement is true about the density f(y) for 0 ≤ y ≤ 2.5?

F(y) = 0.2

F(y) = 2.5

F(y) = 0.4

F(y) = 1/y

f(y) = derivative of F(y) = 0.4

Explanation

f(y) = derivative of F(y) = 0.4

20) For a discrete variable, how does the CDF typically look?

A step function with jumps at the variable’s values

Smooth and curved

A line with negative slope

A symmetric bell curve

Discrete CDF jumps at each value where probability exists

Explanation

Discrete CDF jumps at each value where probability exists