Common distributions of random variables Flashcards
Bernoulli distribution?
x - Bernoulli (π)
π^x (1-π)^1-x
only two outcomes
Mean of Bernoulli?
-π
Variance of Bernoulli?
-π(1-π)
MGF of Bernoulli?
-(1-π) +πe^t
Binomial distribution?
(n) p^x q^n-x
(x)
(n) meaning?
(x)
nCr = n!/(n-r)! r!
Mean of binomial?
-nπ
Variance of Binomial
-nπ(1-π)
Poisson distribution?
- (λ^x e^-λ ) / x!
Mean of Poisson?
λ
Variance of Poisson?
λ
MGF of Poisson?
e^λ(e^t -1)
When it is suitable for a Poisson distribution?
It models the number of events (iceberg calving) occurring in a fixed interval of time.
Events occur independently.
Events occur at a constant average rate.
When can we use Poisson to approximate binomial?
-when n is large and π is small
nπ =λ
Uniform Distribution?
1/(b-a)
Mean and variance of uniform distribution?
- E(x) = a+b /2
-Var(x) = (b-a)^2/12
Exponential distribution?
λe^-λx
X - Exponential (λ)
Exponential distribution mean and variance?
E(x) = 1/λ
Var(x) = 1/λ^2
Median of exponential distrubution?
-ln 2 / λ
How do we use exponential distribution when a Poisson distribution is used?
- use same lambda
-using CDF of exponential for time between intervals of Poisson
CDF exponential for Poisson?
-time is less = 1-e^-λt
-time is more = e^-λt
Normal distribution mean and variance?
-E(x) = μ
-Var(x) = σ^2
mean=median=mode
In normal distribution how to describe Y = aX+b
Y - N( aμ +b, a^2 σ^2)
How do we calculate probability using normal distribution?
-using z value
-(x- μ /σ )
rearrange for P(Z>x) , then 1 - table value
How do we approximate binomial distribution using normal distribution?
-N (nπ , nπ (1-π))
-if n is large
-if nπ >5 and nπ(1-π)>5
When we approximate binomial using normal what do we account for?
-subtract 0.5 for For P(X≥k) and P(k<X)
-add 0.5 for P(X≤k) and P(k>X)
