Exponential Expected Value Calculator

Expected Value Calculator - Exponential Distribution

Calculate the expected value (average wait time) for an exponential distribution. Perfect for modeling time between events: server response times, time until failure, or any memoryless waiting process.

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Key Insight: E(X) = 2.00 is the reciprocal of the rate. With λ = 0.5, you expect 1/0.5 = 2.00 time units between events on average.

Parameter

λ (Rate Parameter)

Formula:E(X) = 1 / λ

Variance:1 / λ²

Memoryless:P(X>s+t | X>s) = P(X>t)

Percentiles

Mode:0.00

Q1 (25%):0.58

Median (50%):1.39

Q3 (75%):2.77

Mean:2.00

Expected Value

E(X) = 2.000

1 / λ = 1 / 0.5

Average time between events

Probability Density Function with E(X) = 2.00

Density

E(X) = 2.00

Median = 1.39

05.010.0

Note: Exponential decay from λ = 0.500 at x=0. Red line shows mean E(X), green shows median (always smaller for exponential). Memoryless property: P(X>s+t | X>s) = P(X>t).

📊Moderate wait times

Average: 2.00 time units between events
50% of waits are less than 1.39 units
Very variable: std dev = 2.00

Understanding Expected Value for Exponential Distribution

What Does E(X) Mean?

The expected value E(X) represents the average waiting time or time between events. For exponential distributions, this is simply 1/λ where λ is the rate of events.

Interpreting Your Result

With E(X) = 2.00, this means:

Average wait time: 2.00 units
Median wait is shorter: 1.39 units
High variability: σ = μ = 2.00

Real-World Examples

Server requests (λ=2/sec): E(X) = 0.5 sec between requests
Equipment failure (λ=0.1/hour): E(X) = 10 hours until failure
Customer arrivals (λ=5/min): E(X) = 0.2 min between customers
Radioactive decay (λ=0.01/year): E(X) = 100 years

Why E(X) = 1 / λ

The exponential distribution models the time until an event occurs when events happen at constant rate λ. If events occur at rate λ per time unit, the average time between events is naturally 1/λ. This reciprocal relationship comes directly from integrating x·f(x) over the exponential PDF.

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.