Download The density function of a normal distribution with mean μ and

The density function of a normal distribution with mean µ and standard deviation σ :
! ! =
!
!!! !
exp −
!!! !
!! !
,
where -∞ < x < ∞, ∞< µ < ∞, 0 < σ <∞. For the standard normal distribution, µ = 0 and σ = 1.
Another distribution that a bona fide statistician cannot live without is the gamma distribution. Its density
function is:
! ! = !
!(!)! !
! !!! ! !!/! where x > 0; α , β > 0.
The gamma distribution includes as special cases the exponential distribution (α = 1) and the chi-square(d)
distribution (α = n/2, β = 2); a limiting case includes the normal distribution (α → ∞).
The uniform distribution on the unit interval has density function
g(x) = 1
0 < x < 1.
If the random variable X has this uniform distribution, then Y = F -1 (X) has distribution function F.
A useful version of the Central Limit Theorem states that given a random sample X1, X2, …, Xn from a
distribution with mean µ and standard deviation σ, the distribution of the sample mean approaches the
normal distribution with mean µ and standard deviation σ / √n . (The random sample wording implies that
the Xi’s are independent and identically distributed; the variance σ 2 is assumed to be finite).
Mean: E(X) = µ
Variance: Var (X) = E[(X-µ)2] = σ 2 (standard deviation σ has same units as X).
Skewness: E[(X-µ)3/σ 3]
(zero for symmetric distributions)
Kurtosis: E[(X-µ)4/σ 4]
(equals 3 for the normal distribution)
The probability mass function of the discrete binomial distribution is given by:
! ! = ! = Note that
!
!
= !
!
!!
!! !!! !
! ! (1 − !)!!!
for k = 0, 1, …, n; 0 < p < 1.
, where n! = n × (n-1) × (n-2) × … × 2 × 1.