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Negative Binomial Expected Value Calculator


Expected Value Calculator - Negative Binomial Distribution

Calculate the expected value (average number of failures before r successes) for a negative binomial distribution. Perfect for quality control, sales quotas, or any scenario where you need a certain number of successes.

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Key Insight: E(X) = 7.00 means on average you will experience 7.00 failures before getting your 3 required successes. Total trials needed: about 10.00.

Parameters

Formula:E(X) = r(1-p) / p
Success rate:30.0%
Expected total trials:10.00
Expected Value (Failures Before 3 Successes)
E(X) = 7.0000
Formula: r(1-p)/p = 3×0.70/0.3 = 7.00
Interpretation: On average, expect 7.00 failures before getting 3 successes
📊E(X) in Context: Moderate success rate
  • 30% success rate, need 3 successes? Expect 7.0 failures
  • Total expected trials: 10.0
  • Typical range: 3 to 12 failures
Probability Distribution with E(X) = 7.00
Probability
E(X) = 7.00
0.027
0
0.057
1
0.079
2
0.093
3
0.097
4
0.095
5
0.089
6
0.080
7
0.070
8
0.060
9
0.050
10
0.042
11
0.034
12
0.027
13
0.022
14
0.017
15
0.014
16
0.011
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Number of Failures

Understanding Expected Value for Negative Binomial Distribution

What Does E(X) Mean?

The expected value E(X) represents the average number of failures you will experience before achieving r successes. It is the cost in terms of failed attempts to reach your success goal.

Interpreting Your Result

With E(X) = 7.00, this means:

  • Expect 7.00 failures before 3 successes
  • Total trials needed: about 10.00
  • Each trial is independent with 30% success rate

Real-World Examples

  • Sales (r=5, p=0.2): Need 5 sales? Expect 20 rejections
  • Quality control (r=3, p=0.05): Find 3 defects? Expect 57 good items
  • Job interviews (r=2, p=0.3): Need 2 offers? Expect 4.7 rejections

Why E(X) = r(1-p) / p

This is a generalization of the geometric distribution. For one success (r=1), you expect (1-p)/p failures. For r successes, you need r times as many failures on average. This assumes each trial is independent with constant probability p.


Understanding Expected Value
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Calculate Expected Value

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



Understanding Expected Value


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