Continuous Uniform Expected Value Calculator

Expected Value Calculator - Continuous Uniform Distribution

Calculate the expected value (average value) for a continuous uniform distribution. Perfect for modeling scenarios where all values in a range are equally likely: random numbers, waiting times, or any situation with equal probability across an interval.

💡

Key Insight: E(X) = 5.00 is simply the midpoint of your range [0, 10]. For uniform distributions, the expected value is always the average of the endpoints: (a + b) / 2.

Parameters

a (Minimum)

b (Maximum)

Formula:E(X) = (a + b) / 2

Range:10.00

Percentiles

Min:0.00

Q1:2.50

Median:5.00

Q3:7.50

Max:10.00

Expected Value

E(X) = 5.000

(0 + 10) / 2

The midpoint of [0, 10]

Probability Density Function with E(X) = 5.00

Density

E(X) = 5.00

a = 0.00E(X) = 5.00b = 10.00

Note: The PDF is constant at height 0.1000 across [0, 10]. Total area under the curve = 1. Shaded region shows E(X) ± σ (≈57.7% of distribution).

📊Small uniform range

Values uniformly distributed from 0 to 10? Expect 5.00 on average
Over 100 samples? Expect total ≈ 500
Middle value 5.00 is both mean and median

Understanding Expected Value for Continuous Uniform Distribution

What Does E(X) Mean?

The expected value E(X) represents the average value of the distribution - the 'center of mass' of the PDF. For uniform distributions, this is simply the midpoint of the interval [a, b].

Interpreting Your Result

With E(X) = 5.00, this means:

The average value across the distribution is 5.00
This equals the median and mode (all the same for uniform)
Values are evenly distributed around this center point

Real-World Examples

Random number [0, 1]: E(X) = 0.5
Arrival time [0, 60] min: E(X) = 30 minutes
Temperature [20, 30]°C: E(X) = 25°C
Duration [5, 15] sec: E(X) = 10 seconds

Why E(X) = (a + b) / 2

The continuous uniform distribution has constant probability density across [a, b]. When you integrate x · f(x), you are finding the weighted average where all weights are equal. This naturally gives you the midpoint. It is the same as averaging two numbers: (a + b) / 2.

Understanding Expected Value

Formula and Calculation

Applications

Calculate Expected Value

Use the calculator below to compute the expected value with step-by-step solutions and detailed explanations.