Discrete Continuous Uniform Quiz: Discrete Vs Continuous Uniform

1) For a discrete uniform distribution over n equally likely outcomes {x1, x2, …, xn}, what is P(X = xi) for any i?

N

1/(b − a)

1/n

(b − a)/n

In a discrete uniform distribution, all n outcomes are equally likely, so each row in the probability table has the same probability mass: P(X = xi) = 1/n. This ensures the column sums to 1 because n·(1/n) = 1.

Explanation

In a discrete uniform distribution, all n outcomes are equally likely, so each row in the probability table has the same probability mass: P(X = xi) = 1/n. This ensures the column sums to 1 because n·(1/n) = 1.

2) For a continuous uniform distribution on the interval [a, b], what is the probability density f(x) on that interval?

1/(b − a) for a ≤ x ≤ b, 0 otherwise

1/n for a ≤ x ≤ b, 0 otherwise

N for all x

(b − a) for a ≤ x ≤ b, 0 otherwise

A continuous uniform distribution spreads probability evenly across the interval [a, b], so its height (density) is constant: f(x) = 1/(b − a) on [a, b] and 0 outside. The total area is ∫_a^b 1/(b − a) dx = (b − a)/(b − a) = 1.

Explanation

A continuous uniform distribution spreads probability evenly across the interval [a, b], so its height (density) is constant: f(x) = 1/(b − a) on [a, b] and 0 outside. The total area is ∫_a^b 1/(b − a) dx = (b − a)/(b − a) = 1.

3) For a fair six‑sided die modeled as discrete uniform on {1,2,3,4,5,6}, P(X = 4) = 1/6.

True

False

There are n = 6 equally likely outcomes and each gets probability 1/n = 1/6 in a discrete uniform table; the row for x = 4 therefore has P(X = 4) = 1/6.

Explanation

There are n = 6 equally likely outcomes and each gets probability 1/n = 1/6 in a discrete uniform table; the row for x = 4 therefore has P(X = 4) = 1/6.

4) For a continuous uniform distribution on [2, 8], the constant density f(x) on that interval is ________.

Density must integrate to 1: f(x)·(8 − 2) = 1 → f(x) = 1/6 for all x in [2, 8].

Explanation

Density must integrate to 1: f(x)·(8 − 2) = 1 → f(x) = 1/6 for all x in [2, 8].

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5) In a discrete uniform distribution on {1,2,3,4,5}, what is P(X ∈ {2,4})?

1/5

2/5

3/5

4/5

Each outcome has probability 1/5. The event {2,4} contains 2 outcomes, so P = 2·(1/5) = 2/5. In table form, you add the two rows’ probabilities.

Explanation

Each outcome has probability 1/5. The event {2,4} contains 2 outcomes, so P = 2·(1/5) = 2/5. In table form, you add the two rows’ probabilities.

6) For X ~ Uniform(0, 10) (continuous), what is P(3 ≤ X ≤ 7)?

0.2

0.3

0.4

0.5

For continuous uniform, probabilities equal interval length divided by total length: (7 − 3)/(10 − 0) = 4/10 = 0.4. Geometrically, area of a rectangle of width 4 and height 1/10.

Explanation

For continuous uniform, probabilities equal interval length divided by total length: (7 − 3)/(10 − 0) = 4/10 = 0.4. Geometrically, area of a rectangle of width 4 and height 1/10.

7) Select all scenarios that are well‑modeled by a discrete uniform distribution (all outcomes equally likely and countable).

Rolling a fair six‑sided die (outcomes {1,…,6})

Choosing a random real number in [0,1]

Picking a random integer from 1 through 10 with equal chance

Measuring a random time uniformly between 0 and 60 seconds

Drawing one winning number uniformly from {1,…,50} in a lottery

Discrete uniform requires a finite or countable set with equal probability to each element. A: 6 faces all equal. C: 10 integers equally likely. E: 50 integers equally likely. B and D are continuous intervals, hence continuous uniform, not discrete.

Explanation

Discrete uniform requires a finite or countable set with equal probability to each element. A: 6 faces all equal. C: 10 integers equally likely. E: 50 integers equally likely. B and D are continuous intervals, hence continuous uniform, not discrete.

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8) Which statement best distinguishes discrete uniform from continuous uniform?

They are identical except for notation

Discrete uniform allows negative probabilities; continuous does not

Continuous uniform has unequal density; discrete uniform has unequal masses

In discrete uniform the table lists equal masses (each row gets 1/n). In continuous uniform the table is replaced by a density function that is constant on [a,b] and P(X = exact value) = 0, while probabilities come from interval lengths.

Explanation

In discrete uniform the table lists equal masses (each row gets 1/n). In continuous uniform the table is replaced by a density function that is constant on [a,b] and P(X = exact value) = 0, while probabilities come from interval lengths.

9) For a continuous uniform distribution on [a, b], P(X = a) = 0.

True

False

In continuous distributions, exact points have zero probability because probability is area under a curve over intervals. A single point has zero width; hence P(X = a) = ∫_a^a f(x) dx = 0.

Explanation

In continuous distributions, exact points have zero probability because probability is area under a curve over intervals. A single point has zero width; hence P(X = a) = ∫_a^a f(x) dx = 0.

10) What is the mean (expected value) of a continuous uniform distribution on [a, b]?

B − a

A + b

(a + b)/2

(b − a)/2

The uniform density is symmetric on [a, b], so the balance point is the midpoint: E[X] = ∫_a^b x·(1/(b − a)) dx = (a + b)/2.

Explanation

The uniform density is symmetric on [a, b], so the balance point is the midpoint: E[X] = ∫_a^b x·(1/(b − a)) dx = (a + b)/2.

11) What is the variance of a continuous uniform distribution on [a, b]?

(b − a)^2/12

(b − a)^2/6

(b − a)/12

(b − a)^2/4

Var(X) = ∫_a^b (x − (a + b)/2)^2 · (1/(b − a)) dx = (b − a)^2/12, a standard result derived by evaluating the second moment about the mean.

Explanation

Var(X) = ∫_a^b (x − (a + b)/2)^2 · (1/(b − a)) dx = (b − a)^2/12, a standard result derived by evaluating the second moment about the mean.

12) For a discrete uniform distribution on {1,2,…,n}, what is the mean E[X]?

(n + 1)/2

N/2

N + 1

1/n

The average of the first n equally spaced integers is the midpoint: (1 + n)/2. This also follows from symmetry of the equally likely outcomes.

Explanation

The average of the first n equally spaced integers is the midpoint: (1 + n)/2. This also follows from symmetry of the equally likely outcomes.

13) For a discrete uniform distribution on {1,2,…,n}, what is the variance Var(X)?

(n^2 − 1)/12

N^2/12

(n − 1)^2/12

(n^2 + 1)/12

A known result for discrete uniform integers 1..n is Var(X) = (n^2 − 1)/12, derivable from E[X^2] = (n(n + 1)(2n + 1))/6n and E[X] = (n + 1)/2.

Explanation

A known result for discrete uniform integers 1..n is Var(X) = (n^2 − 1)/12, derivable from E[X^2] = (n(n + 1)(2n + 1))/6n and E[X] = (n + 1)/2.

14) For a discrete uniform distribution on {10, 20, 30, 40}, P(X = 20) = ________.

There are 4 equally likely outcomes; each has probability 1/4 = 0.25. In the probability table, each row’s P(X = x) equals 0.25 and the column sums to 1.

Explanation

There are 4 equally likely outcomes; each has probability 1/4 = 0.25. In the probability table, each row’s P(X = x) equals 0.25 and the column sums to 1.

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15) For X ~ Uniform(5, 9) (continuous), what is P(X < 7)?

0.25

0.4

0.5

0.75

Probability is the length of the interval divided by total length: (7 − 5)/(9 − 5) = 2/4 = 0.5. Graphically, it is area of a rectangle of width 2 and height 1/4.

Explanation

Probability is the length of the interval divided by total length: (7 − 5)/(9 − 5) = 2/4 = 0.5. Graphically, it is area of a rectangle of width 2 and height 1/4.

16) Which sets can be supports of a discrete uniform distribution? Select all that apply.

{A, B, C} with equal chance

Integers 1 through 6 with equal chance

All real numbers in [0, 1]

Odd integers from 1 through 9 with equal chance

All real numbers

Discrete uniform requires a finite or countable set of distinct outcomes with equal mass per element: (A) three labels, (B) six integers, (D) five odd integers. The real intervals (C,E) are uncountable and model continuous uniform instead.

Explanation

Discrete uniform requires a finite or countable set of distinct outcomes with equal mass per element: (A) three labels, (B) six integers, (D) five odd integers. The real intervals (C,E) are uncountable and model continuous uniform instead.

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17) In a discrete uniform distribution with 8 equally likely outcomes, each outcome has probability 0.125.

True

False

Each outcome has probability 1/n. With n = 8, P = 1/8 = 0.125 for every row, and 8 × 0.125 = 1 verifies the sum constraint.

Explanation

Each outcome has probability 1/n. With n = 8, P = 1/8 = 0.125 for every row, and 8 × 0.125 = 1 verifies the sum constraint.

18) For X ~ Uniform(a, b) (continuous) and a ≤ c ≤ d ≤ b, which formula gives P(c ≤ X ≤ d)?

(b − a)/(d − c)

(d − c)/(b − a)

1/(d − c)

C/d

Under a constant density 1/(b − a), probability over [c, d] equals density × interval length: (1/(b − a))·(d − c) = (d − c)/(b − a).

Explanation

Under a constant density 1/(b − a), probability over [c, d] equals density × interval length: (1/(b − a))·(d − c) = (d − c)/(b − a).

19) Which scenario is best modeled as a continuous uniform distribution?

Drawing one card from 4 equally likely suits

Rolling a fair six‑sided die

Choosing a random integer from 1 to 10 with equal chance

Selecting a random real number between 0 and 100 with equal likelihood for every value

Continuous uniform applies to uncountably many outcomes spread evenly across an interval. Real numbers on [0,100] fit this; the others are discrete, countable sets.

Explanation

Continuous uniform applies to uncountably many outcomes spread evenly across an interval. Real numbers on [0,100] fit this; the others are discrete, countable sets.

20) In a continuous uniform model on [a, b], P(X = c) = 0 but P(c − ε ≤ X ≤ c + ε) > 0 for any small ε > 0 within [a, b].

True

False

Single points have zero probability, but any non‑zero interval around c has positive length; its probability equals interval length divided by (b − a).

Explanation

Single points have zero probability, but any non‑zero interval around c has positive length; its probability equals interval length divided by (b − a).

21) For a continuous uniform distribution on [−3, 3], the constant density f(x) on the interval is ________.

Total length is b − a = 3 − (−3) = 6. The density must integrate to 1, so f(x) = 1/6 for all x in [−3, 3] and 0 outside.

Explanation

Total length is b − a = 3 − (−3) = 6. The density must integrate to 1, so f(x) = 1/6 for all x in [−3, 3] and 0 outside.

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Discrete Continuous Uniform Quiz: Discrete vs Continuous Uniform