Continuous Probability Distributions Flashcards
Uniform distribution definition
All values over a given range have equal probability density
Cont. Uniform parameters
𝑋~𝑈 (𝛼, 𝛽)
where X is defined over (𝛼, 𝛽) in a sample space
cont. exponential distribution definition
X=time until next event occurs/time between events.
continuous equivalent to geometric
Exponential assumptions
> Events occur randomly at constant rate λ
Events in non-overlapping intervals are independent
Events occur uniformly
Normal distribution parameters
X~N(𝜇, 𝜎2)
Z transformation
Z = (X - 𝜇) / 𝜎
>be wary of using sd above and not variance
Phi(z) use
used to refer to the value in z table corresponding to that value of z
central limit theorem
> where n is large (>30)
distribution will tend towards normal
X ~ N(𝜇, 𝜎2/n)
using mean and variance of original distribution.
Normal approximation to binomial
> n must be large
sum of n independent bernoulli trials with success probability p (mean p, variance pq)
𝑋~𝑁(𝑛𝑝, 𝑛𝑝𝑞)
Normal approximation to Poisson
> rate parameter 𝜆𝑡
(a𝑣𝑒𝑟𝑎𝑔𝑒 𝑛𝑜. 𝑜𝑓 𝑒𝑣𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡𝑖𝑚𝑒 𝑝𝑒𝑟𝑖𝑜𝑑 𝑡)
Sum of t independent Poisson random variables with rate 𝜆 (mean 𝜆 and variance 𝜆)
𝑋~𝑁(𝜆𝑡, 𝜆𝑡) approximately
normal approximation requirements
> Binomial: np >= 5 and nq >= 5
Poisson: 𝜆 >10
𝐸 [𝑋 +- 𝑌] (for 2 random variables with joint distributions)
E[X] +/- E[Y]
𝐸 [𝑋𝑌] for 2 independent random variables
𝐸 [𝑋] * 𝐸 [𝑌]
var[𝑋 +/- 𝑌] for 2 independent random variables
var[𝑋] + var[𝑌]
Remember when using the exponential distribution
to convert the units of x into those of the rate parameter (lambda)
