Probability Distributions Flashcards
When is a Bernoulli distribution used?
Whenever only two outcomes are
possible: one is identified with success and the other with failure.
What other distributions are formed using bernouill trials
Binomial
Geometric
Negative binomial
How is the sum of bernouilli iid RVs distributed
Binomial(n,p)
What is a poisson random variable usually used for
Time between events. Probability of success in poisson distribution is usually Quite small - used for rare events
What does the poisson limit theorem state
X is approximately distributed as poisson p for X~Bin(n,p/n) as n goes to infinity. So pmf of X tends to poisson pmf as n goes to infinity
What are two ways to show an RV is identically distributed
Shows cdfs are the same or mgfs are the same
What practical application can we take from the poisson limit theorem
For very big n instead of binomial we can use poisson distribution
Explain the continuous unfirom distribution
The continuous uniform distribution is defined by spreading mass
uniformly over an interval [a, b]
Explain the gamma family
Continuous distributions for outcomes in [0,infinity)
If n is an integer what si the value of gamma(n)
n-1 !
What is the relationship between gamma and exponential
Gamma(1, lamda) is an exponential(lamda) RV
What is the relationship between gamma and chi squared
Gamma (v/2,1/2) is a chi squared X^2(v) distribution
What is the relationship between weibull and exponential
If X ~ Exp(1)
W is weibull (theta, lamda) where W=(X^(1/theta))/lamda
Why is a normal distribution one of the most used distributions
It occurs naturally in a variety of contexts.
- It is analytically tractable.
- It has the familiar bell shape with symmetry property
- Its parameters
- CLT
What distributions can use Chebyshev and markov inequality
They make no assumptions about distribution. Used on any distribution
