Answer
Success!
Kurtosis = 0.03125
↓Steps Explained:↓
A binomial distribution has a probability of success = 0.8
Calculate the probability of exactly 3 successes in 8 trials:
The binomial probability formula is as follows:
| f(k;n,p) = | n! * pkqn - k | |
| k!(n - k)! |
Calculate q - the probability of failure:
q = 1 - p
q = 1 - 0.8
q = 0.2
Calculate n!:
n! = 8!
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
8! = 40320
Calculate k!:
k! = 3!
3! = 3 * 2 * 1
3! = 6
Calculate (n - k)!:
(n - k)! = (8 - 3)!
(n - k)! = 5!
5! = 5 * 4 * 3 * 2 * 1
5! = 120
Calculate the binomial distribution probability:
| P(X = 3) = | 8! * 0.830.2(8 - 3) | |
| 3!(8 - 3)! |
| P(X = 3) = | 40320 * 0.512 * 0.25 | |
| 6 * 120 |
| P(X = 3) = | 40320 * 0.512 * 0.00032 | |
| 720 |
| P(X = 3) = | 6.6060288 | |
| 720 |
Calculate nq to see if we can use the Normal Approximation:
Since q = 1 - p, we have n(1 - p) = 8(1 - 0.8)
nq = 8(0.2)
nq = 1.6
Calculate the mean μ (expected value)
μ = np
μ = 8 x 0.8
μ = 6.4
Calculate the variance σ2
σ2 = np(1 - p)
σ2 = 8 x 0.8 x (1 - 0.8)
σ2 = 6.4 x 0.2
σ2 = 1.28
Calculate the standard deviation σ
σ = √σ2 = √np(1 - p)
σ = √1.28
σ = 1.1314
Calculate Skewness:
| Skewness = | 1 - 2p |
| √np(1 - p) |
| Skewness = | 1 - 2(0.8) |
| √8(0.8)(1 - 0.8) |
| Skewness = | 1 - 1.6) |
| √8(0.8)(0.2) |
| Skewness = | -0.6 |
| √1.28 |
Skewness = -0.46875
Calculate Kurtosis:
| Kurtosis = | 1 - 6p(1 - p) |
| np(1 - p) |
| Kurtosis = | 1 - 6(0.8)(1 - 0.8) |
| 8(0.8)(1 - 0.8) |
| Kurtosis = | 1 - (4.8)(0.2) |
| 8(0.8)(0.2) |
| Kurtosis = | 1 - 0.96 |
| 1.28 |
Kurtosis = 0.03125
Final Answer
Kurtosis = 0.03125Take the Quiz Switch to Chat Mode
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