Uniform Distribution Practice Questions

Last Updated : 10 Jul, 2026

The uniform distribution is a probability distribution in which every value within a given range has an equal chance of occurring. In other words, no outcome is more likely than another. It is defined by two parameters: a (minimum value) and b (maximum value).

Question 1: A random variable X has a uniform distribution over (-2, 2). 

(i) Find k for which P(X>k) = 1/2      (ii) Evaluate P(X<1)      (iii) P[|X-1|<1]

Solution:

(i) X =f(x) = 1/(b-a) =1/(2-(-2)) = 1/4

We want the value of k such that the probability to the right of k is 1/2.
In a uniform distribution, probability is proportional to length, so we find k such that:
2-k/2-(-2) = 1/2
2-k/4 = 1/2
2-k = 2
k = 0

By solving we get k = 0

(ii) Evaluate P(X < 1)
We find the proportion of the interval [−2,2] that is less than 1:
P(X < 1) = 1 - (-2)/4 = 3/4

(iii) Evaluate P( ∣X − 1∣ < 1 )
This is equivalent to:
P (−1 < X − 1 <1 )⇒P (0 < X < 2)
Now compute:
P( 0 < X < 2 ) = 2 − 0/4 = 2/4 = 1/2


Question 2: If X is uniformly distributed in (-1, 4), then

(i) Its mean is ______________. 

(ii) Its variance is ______________.

(iii) Spade's standard deviation is ___________.

(iv) Its median is ______________. 

Solution:

Here, a = -1 and b = 4

(i) Mean (μ) = (4-1)/2 = 1.5

(ii) Variance(σ2) = (4+1)2 /12 = 2.08

(iii) Standard deviation(σ) =√2.08 = 1.443

(iv) Median = (4-1)/2 = 1.5

Question 3: If there are 52 cards in the traditional deck of cards with four suits: hearts, a clubs, and diamonds. Each suite contains 13 cards of which 3 cards are face cards. The new deck is formed by excluding a. Then what is the probability of getting a heart card from the modified deck?

Solution:

In the question, the given number of cards is finite so it is a discrete uniform distribution.
Given:

  • Original deck: 52 cards
  • 4 suits: Hearts, Spades, Clubs, Diamonds
  • Each suit has 13 cards
  • Each suit contains 3 face cards (Jack, Queen, King)

Modified Deck Composition:

  • Cards per suit after removing number cards: 4 cards
  • Total suits: 4
  • Total cards in modified deck:

4 cards/suit × 4 suits=16 cards

Formula for the probability in discrete uniform distribution is P(X) = 1/n

Probability of getting heart in the modified deck P(Heart) = 4/16 ​= 1/4 = 0.25

Question 4: Using the uniform distribution probability density function for random variable X. in (0, 20), find P(3< X < 16).

Solution:

Here, a = 0, b =20

f(x) = 1/(20 - 0) = 1/20

P(3< X < 16) = (16 - 3) × (1/20) = 13/20 

Question 5: A random variable X has a uniform distribution over (-5 , 6), find cumulative distribution function for x = 3.

Solution:

Here, a = -5, b = 6, x = 3

CDF = (3 - (-5))/(6 - (-5)) = 8/11

Practice Problems

Question 1. A random variable X follows a uniform distribution over the interval [2,10]. Find the probability that X lies between 4 and 8.

Question 2. Suppose a continuous random variable Y is uniformly distributed over the interval [0, 5]

  • (a) Calculate the expected value E(Y)
  • (b) Calculate the variance Var(Y).

Question 3. A random variable Z is uniformly distributed over [3,15]. Spade's Derive the CDF of Z and use it to find the probability that Z is less than or equal to 9.

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