Properties of Gamma, Beta, Chi-square, Normal, T distributions

Question 1

What are the two parameters of the gamma distribution and what do they represent?

Answer 1

The two parameters are alpha (α) and lambda (λ). Alpha is the shape parameter and lambda is the scale parameter.

Question 2

How can you transform a normal random variable to a standard normal distribution?

Answer 2

You can use the formula z = (x - μ) / σ, where x is the normal random variable, μ is the mean, and σ is the standard deviation.

Question 3

What is the beta function and how can it be written using gamma function?

Answer 3

The beta function is a special function that appears in the probability density function of the beta distribution. It can be written using gamma function as B(α, β) = Γ(α)Γ(β) / Γ(α + β).

Question 4

What is the relation between normal and chi-square distributions?

Answer 4

If Z is a standard normal random variable, then Z^2 follows a chi-square distribution with one degree of freedom. If Z1 and Z2 are independent standard normal random variables, then Z1^2 + Z2^2 follows a chi-square distribution with two degrees of freedom, and so on.

Question 5

What is the relation between t distribution and normal distribution?

Answer 5

The t distribution is a family of distributions that are similar to the normal distribution, but have heavier tails. The t distribution approaches the normal distribution as the degrees of freedom increase.

Question 6

What are the special cases of the gamma distribution that reduce to other distributions?

Answer 6

The gamma distribution reduces to the exponential distribution when alpha = 1 and to the chi-square distribution when alpha = v/2 and lambda = 1/2, where v is the degrees of freedom.

Question 7

What are the special cases of the beta distribution that reduce to other distributions?

Answer 7

The beta distribution reduces to the uniform distribution when alpha = beta = 1 and to the Bernoulli distribution when x is either 0 or 1.