Understanding Cumulative Distribution Functions Quiz

1) 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

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) 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

4) 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

5) 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

6) 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

7) 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

8) 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)

9) 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

10) 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 < X ≤ b) = F(b) − F(a)

Explanation

P(a < X ≤ b) = F(b) − F(a)

11) 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

12) 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

13) 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

14) 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

15) 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

16) 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

17) 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 < X ≤ 3) = F(3) − F(1)

Explanation

P(1 < X ≤ 3) = F(3) − F(1)

18) 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

19) 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

20) 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

Understanding Cumulative Distribution Functions Quiz

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

2)

What first name or nickname would you like us to use?

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

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

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

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

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

7) What is P(Y > 2.5)?

8) What is P(1 < Y ≤ 2)?

9) What is P(Y ≤ 1)?

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

11) What is F(0)?

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

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

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

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

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

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

18) What is F(3)?

19) What is F(2)?

20) What is F(1)?