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Answer

↓Steps Explained:↓

A binomial distribution has a probability of success = 0.4

Calculate the probability of exactly 3 successes in 10 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 - 0.4

q = 0.6

Calculate n!:

n! = 10!

10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

10! = 3628800

Calculate k!:

k! = 3!

3! = 3 * 2 * 1

3! = 6

Calculate (n - k)!:

(n - k)! = (10 - 3)!

(n - k)! = 7!

7! = 7 * 6 * 5 * 4 * 3 * 2 * 1

7! = 5040

Calculate the binomial distribution probability:

P(X = 3)  =	10! * 0.430.6(10 - 3)
	3!(10 - 3)!

P(X = 3)  =	3628800 * 0.064 * 0.67
	6 * 5040

P(X = 3)  =	3628800 * 0.064 * 0.0279936
	30240

P(X = 3)  =	6501.32324352
	30240

Calculate nq to see if we can use the Normal Approximation:

Since q = 1 - p, we have n(1 - p) = 10(1 - 0.4)

nq = 10(0.6)

nq = 6

Calculate the mean μ (expected value)

μ  =  np

μ  =  10 x 0.4

μ = 4

Calculate the variance σ²

σ²  =  np(1 - p)

σ²  =  10 x 0.4 x (1 - 0.4)

σ²  =  4 x 0.6

σ² = 2.4

Calculate the standard deviation σ

σ  =  √σ² = √np(1 - p)

σ  =  √2.4

σ = 1.5492

Calculate Skewness:

Skewness = 0.083333333333333

Calculate Kurtosis:

Kurtosis = -0.18333333333333

Final Answer