Discrete Distributions

Question 1

What is negative binomial distribution?

Answer 1

Negative binomial distribution is a distribution of the fixed number of trials required to obtain r successes, where r > 2

Question 2

What is the formula for probability for negative binomial?

Answer 2

Here

Where x = number of failures (i.e. number of trials minus number of success), r = number of successes, p = probability of one success.

Question 3

What is the formula for conditional density function of x?

Answer 3

f(x|y) = f(x, y)/f(y) , where f(y) is the marginal density function

Question 4

What is the formula for conditional density function of y?

Answer 4

f(y|x) = f(x, y)/f(x) , where f(x) is the marginal density function of x

Question 5

What is Baye’s Theorem?

Answer 5

Here

Question 6

What is the formula for moment about the origin?

Answer 6

(Σx^r)/N

N = number of terms, r = whether it be first, second, third, etc, moment.

Question 7

What is the formula for moment about the mean?

Answer 7

Σ(x - x̄)^r/N
Where x̄ = mean of values,
N = number of terms
r = first, second, etc, moment

Question 8

What is the formula for moment about an arbitrary value?

Answer 8

Σ(x - b)^r/N
b = arbitrary value

Question 9

What are the properties of moments?

Answer 9

If m” = moment about the mean, m^k = moment about arbitrary value, m = moment about origin, then:

1. m”₂ = m₂ - (m₁)²
2. If x̄ = b = 0:
   2.a. m₂ = m”₂ = m^k ₂, and
   2.b. m₁ = m”₁ = m^k ₁
3. m₁ = m”₁ if x̄ = b
   Where b is an arbitrary value.

Question 10

What are the properties of probability density function for continuous random variable?

Answer 10

Here

Question 11

What is the formula for hypergeometric distribution?

Answer 11

This.

Where n = number of the selected/sample size
N = total size out of which n is selected
a = number of success in N/population
x = number of success in n (sample size)

Question 12

What is the formula for expectation in hypergeometric distribution?

Answer 12

E(X) = n • (a/N)
Where n = number of the selected/sample size
N = total size out of which n is selected
a = number of success in N/population
x = number of success in n (sample size)

Question 13

What is the expectation of binomial distribution?

Answer 13

E(X) = np,
p = probability of event occuring,
n = sample size

Question 14

What is the key factor of Hypergeometric distribution?

Answer 14

It is a distribution whose events are without replacement, meaning they are not independent of one another.

Question 15

What is the formula for probability of a Poisson distribution?

Answer 15

P(x) = λ^X • (e^–λ / X!)
λ = mean
X = amount we are checking the probability for

Question 16

What is the expectation for Poisson distribution?

Answer 16

E(X) = λ

Question 17

What is the variance for Poisson distribution?

Answer 17

Var(X) = λ,
λ = mean

Question 18

When can Poisson distribution be used to approximate binomial?

Answer 18

When n —> ∞ and P —> 0