Uniform Distribution PDF Quiz: Probability Density Function (Uniform Distribution)

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| Questions: 20 | Updated: Dec 16, 2025

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1) For a continuous uniform distribution on [a, b], what is the correct probability density function (PDF) f(x)?

1/(b - a) for a ≤ x ≤ b and 0 otherwise

1/n for a ≤ x ≤ b and 0 otherwise

(b - a) for a ≤ x ≤ b and 0 otherwise

N for all real x

A continuous uniform distribution spreads total probability 1 evenly across the interval [a, b]. The constant density that integrates to 1 is f(x) = 1/(b - a) on [a, b] and 0 outside. The area check is ∫_a^b 1/(b - a) dx = (b - a)/(b - a) = 1.

Explanation

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About This Quiz

Uniform Distribution Pdf Quiz: Probability Density Function (Uniform Distribution) - Quiz

Ever wondered what a PDF looks like for a uniform distribution? This quiz shows how probability spreads evenly across an interval and what the density function tells you. You’ll read graphs, interpret heights, and understand how total probability stays the same. Work through the questions and see how neat and... see morepredictable uniform PDFs really are.
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2) For X ~ Uniform(2, 8), what is the constant density f(x) on the interval?

1/4

1/5

1/6

1/8

The interval length is b - a = 8 - 2 = 6. The uniform density is f(x) = 1/(b - a) = 1/6 for all x in [2, 8], and 0 otherwise. The total probability is area = 6 × (1/6) = 1.

Explanation

The interval length is b - a = 8 - 2 = 6. The uniform density is f(x) = 1/(b - a) = 1/6 for all x in [2, 8], and 0 otherwise. The total probability is area = 6 × (1/6) = 1.

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3) For a valid continuous uniform PDF on [a, b], the integral of f(x) over [a, b] equals 1.

True

False

Total probability under a PDF must be 1. For the uniform PDF f(x) = 1/(b - a) on [a, b], we have ∫_a^b f(x) dx = ∫_a^b 1/(b - a) dx = (b - a)/(b - a) = 1.

Explanation

Total probability under a PDF must be 1. For the uniform PDF f(x) = 1/(b - a) on [a, b], we have ∫_a^b f(x) dx = ∫_a^b 1/(b - a) dx = (b - a)/(b - a) = 1.

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4) For X ~ Uniform(0, 5), the constant density on its interval is ________.

The length of the interval is 5 − 0 = 5, so f(x) = 1/(5) = 1/5 for x in [0, 5]. This integrates to 1 over [0, 5].

Explanation

The length of the interval is 5 − 0 = 5, so f(x) = 1/(5) = 1/5 for x in [0, 5]. This integrates to 1 over [0, 5].

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5) For X ~ Uniform(a, b) and a ≤ c ≤ d ≤ b, what is P(c ≤ X ≤ d)?

(d − c)/(b − a)

(b − a)/(d − c)

1/(d − c)

C/d

Uniform probability over an interval equals density × interval length. With f(x)=1/(b−a), P(c ≤ X ≤ d)=∫_c^d 1/(b−a) dx = (d − c)/(b − a).

Explanation

Uniform probability over an interval equals density × interval length. With f(x)=1/(b−a), P(c ≤ X ≤ d)=∫_c^d 1/(b−a) dx = (d − c)/(b − a).

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6) For X ~ Uniform(3, 9), what is P(4 ≤ X ≤ 7)?

0.3

0.4

0.5

0.6

Compute using length ratio: (7 − 4)/(9 − 3) = 3/6 = 0.5. Geometrically, area of a rectangle with width 3 and height 1/6.

Explanation

Compute using length ratio: (7 − 4)/(9 − 3) = 3/6 = 0.5. Geometrically, area of a rectangle with width 3 and height 1/6.

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7) For a continuous uniform model, what is P(X = c) for any specific c in [a, b]?

1/(b − a)

Depends on c

In continuous distributions, the probability of any single exact point is zero, because intervals of zero width have zero area under the density curve. Probabilities are assigned to intervals, not points.

Explanation

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8) Which of the following are valid PDFs for a uniform distribution on [0, 4]? Select all that apply.

F(x) = 1/4 for 0 ≤ x ≤ 4; f(x) = 0 otherwise

F(x) = 1/5 for 0 ≤ x ≤ 4; 0 otherwise

F(x) = 0.25 for 0 ≤ x ≤ 4; 0 otherwise

F(x) = 1/2 for 0 ≤ x ≤ 4; 0 otherwise

F(x) = 1/4 for 0 < x < 4; 0 otherwise

A and C are the same constant (1/4) on [0,4], integrating to (4 − 0) × 1/4 = 1. E is also valid since endpoints contribute zero measure; ∫_(0)^(4) 1/4 dx = 1. B integrates to 4 × 1/5 = 0.8 (invalid). D integrates to 4 × 1/2 = 2 (invalid).

Explanation

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9) What is the mean (expected value) of X ~ Uniform(a, b)?

A+b

(a + b)/2

B − a

(b − a)/2

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

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10) What is the variance of X ~ Uniform(a, b)?

(b − a)^2/12

(b − a)^2/6

(b − a)/12

(b − a)^2/4

Using Var(X)=E[X^2]−(E[X])^2, with E[X]=(a+b)/2 and E[X^2]=∫_a^b x^2 · (1/(b − a)) dx = (b^3 − a^3)/(3(b − a)), we get Var(X)=(b − a)^2/12.

Explanation

Using Var(X)=E[X^2]−(E[X])^2, with E[X]=(a+b)/2 and E[X^2]=∫_a^b x^2 · (1/(b − a)) dx = (b^3 − a^3)/(3(b − a)), we get Var(X)=(b − a)^2/12.

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11) For a continuous uniform distribution on [a, b], the height of the PDF is the same at every x in [a, b].

True

False

By definition, a continuous uniform distribution has constant density on its support: f(x)=1/(b−a) for every x in [a, b].

Explanation

By definition, a continuous uniform distribution has constant density on its support: f(x)=1/(b−a) for every x in [a, b].

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12) For X ~ Uniform(−3, 3), the constant density on its interval is ________.

The interval length is 3 − (−3) = 6, so the uniform density is f(x) = 1/6 on [−3, 3].

Explanation

The interval length is 3 − (−3) = 6, so the uniform density is f(x) = 1/6 on [−3, 3].

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13) For X ~ Uniform(5, 11), which constant c makes f(x) = c on [5, 11] a valid PDF?

1/5

1/6

1/7

The interval length is 11 − 5 = 6. The density must satisfy c × 6 = 1, so c = 1/6. This ensures ∫_5^11 c dx = 1.

Explanation

The interval length is 11 − 5 = 6. The density must satisfy c × 6 = 1, so c = 1/6. This ensures ∫_5^11 c dx = 1.

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14) For X ~ Uniform(1, 4), what is P(X < 2.5)?

0.3

0.4

0.5

0.6

Compute probability by length ratio: (2.5 − 1)/(4 − 1) = 1.5/3 = 0.5. This is the area of the rectangle with width 1.5 and height 1/3.

Explanation

Compute probability by length ratio: (2.5 − 1)/(4 − 1) = 1.5/3 = 0.5. This is the area of the rectangle with width 1.5 and height 1/3.

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15) Which statements are true for a continuous uniform PDF on [a, b]? Select all that apply.

F(x) is constant on [a, b]

P(X = a) = 0

Total area under f on [a, b] equals 1

Probabilities of subintervals are proportional to their lengths

F(x) increases linearly from a to b

Uniform means constant density: f(x)=1/(b−a). Single points have zero probability, the total area is 1, and any interval’s probability equals its length divided by (b − a). E is false because the density is not increasing; it is constant.

Explanation

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16) For X ~ Uniform(2, 5), the PDF value is 1/3 for all x with 2 ≤ x ≤ 5.

True

False

The interval length is 5 − 2 = 3, so f(x) = 1/(b − a) = 1/3 for x in [2, 5], and 0 otherwise.

Explanation

The interval length is 5 − 2 = 3, so f(x) = 1/(b − a) = 1/3 for x in [2, 5], and 0 otherwise.

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17) For X ~ Uniform(0, 1), what is P(0.2 < X ≤ 0.2)?

0.2

0.5

Any single point has probability 0 in a continuous distribution. The interval (0.2, 0.2] has zero width, so its probability is 0.

Explanation

Any single point has probability 0 in a continuous distribution. The interval (0.2, 0.2] has zero width, so its probability is 0.

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18) Complete the general formula: for X ~ Uniform(a, b), f(x) = ________ for a ≤ x ≤ b and 0 otherwise.

The defining PDF for a continuous uniform distribution over [a, b] is the constant 1 divided by the interval length (b − a).

Explanation

The defining PDF for a continuous uniform distribution over [a, b] is the constant 1 divided by the interval length (b − a).

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19) For X ~ Uniform(0, 20), what is the value of the constant density?

0.02

0.05

0.2

0.5

The interval length is 20 − 0 = 20, so f(x) = 1/20 = 0.05 for x in [0, 20]. The total probability integrates to 20 × 0.05 = 1.

Explanation

The interval length is 20 − 0 = 20, so f(x) = 1/20 = 0.05 for x in [0, 20]. The total probability integrates to 20 × 0.05 = 1.

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20) For X ~ Uniform(−1, 3), what is P(0 ≤ X ≤ 2)?

0.25

0.4

0.5

0.75

Compute by length ratio: total length = 3 − (−1) = 4; subinterval length = 2 − 0 = 2; probability = 2/4 = 0.5.

Explanation

Compute by length ratio: total length = 3 − (−1) = 4; subinterval length = 2 − 0 = 2; probability = 2/4 = 0.5.

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For a continuous uniform distribution on [a, b], what is the correct...

For X ~ Uniform(2, 8), what is the constant density f(x) on the...

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For X ~ Uniform(a, b) and a ≤ c ≤ d ≤ b, what is P(c ≤ X ≤...

For X ~ Uniform(3, 9), what is P(4 ≤ X ≤ 7)?

For a continuous uniform model, what is P(X = c) for any specific c in...

Which of the following are valid PDFs for a uniform distribution on...

What is the mean (expected value) of X ~ Uniform(a, b)?

What is the variance of X ~ Uniform(a, b)?

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For X ~ Uniform(−3, 3), the constant density on its interval is...

For X ~ Uniform(5, 11), which constant c makes f(x) = c on [5, 11] a...

For X ~ Uniform(1, 4), what is P(X < 2.5)?

Which statements are true for a continuous uniform PDF on [a, b]?...

For X ~ Uniform(2, 5), the PDF value is 1/3 for all x with 2 ≤ x ≤...

For X ~ Uniform(0, 1), what is P(0.2 < X ≤ 0.2)?

Complete the general formula: for X ~ Uniform(a, b), f(x) = ________...

For X ~ Uniform(0, 20), what is the value of the constant density?

For X ~ Uniform(−1, 3), what is P(0 ≤ X ≤ 2)?