S2 - Chapters 1-5 Flashcards
Binomial Distribution Function
X ~ B(n, p)
n = number of successes
p = probability of success
Binomial Distribution Conditions
1) FIXED number of trials
2) INDEPENDENT trials
3) CONSTANT probability of success (in each trial)
4) TWO outcomes in each trail - “success” and “failure”
Binomial P(X=x) =
(n choose X) x P^X x (1-p)^n-X
Factorial Function n!
gives us the number of ways of arranging distinguishable objects
Choose Function
(n choose r) = n! / (r! (n-r)! )
Cumulative Distribution Function notation
F(x) = P(X
Cumulative Distribution Function meaning
the probability up to a particular value
Binomial Distribution Mean
E(X) = np
Binomial Distribution Variance
o^2 = np(1-p)
Binomial Distribution Description
The number of “successes” out of n trials, each with a probability of p of success
Poisson Distribution Description
a distribution over the number of events which occur within a period of time, give an average rate lambda
Poisson P(X=x) =
e^ - lambda x lambda^x / x!
Poisson Distribution Conditions
Events must occur:
1) SINGLY in time
2) INDEPENDENTLY of each other
3) at a CONSTANT AVERAGE RATE
Poisson Distribution Mean
E(X) = lambda
Poisson Distribution Variance
o^2 = lambda
Approximating a Binomial using a Poisson
If X ~ B(n,p) and:
- n is large
- p is small
The X can be approximated by X ~ Po(np)
Poisson Distribution Function
X ~ Po (lambda)
Types of discrete random variables
1) Binomial distribution
2) Poisson distribution
3) Discrete uniform distribution
Types of continuous random variables
normal distributions
Probability Density Function (p.d.f)
f(x)
If X is a continuous random variable with p.d.f. f(x) then:
- f(x) >- 0
- P(a
Cumulative Distribution Function
F(x) = P(X
F(x) to f(x)
f(x) = differentiate F(x)
f(x) Mean (if continuous)
E(x) = integrate f(x) times x
f(x) Variance (if continuous)
o^2 = E(X^2) - E(X)^2 o^2 = integrate f(x) times x^2 - mean^2
Mean of F(x)
F(m) = 0.5
- find the range where the median x value is in and then set the function equal to 0.5
Lower Quartile of F(x)
F(Q1) = 0.25
Upper Quartile of F(x)
F(Q3) = 0.75
Positive Skew
mode < median < mean
Negative Skew
mode > median > mean
Continuous Uniform Distribution f(x)
basically a straight horizontal line running from the x value range and you find probability by finding the area under the line.
- the y line is 1 / (b-a)
Continuous Uniform Distribution Mean
E(x) = a + b / 2
Continuous Uniform Distribution Variance
(b-a)^2 / 12
E(X^2) =
Var(X) + E(X)^2
- because Var(X) = E(X^2) - E(X)^2
Continuous Uniform Distribution F(x)
integrate 1 / (b-a) which turns into x-a / b-a
Continuity Corrections
- approximate a discrete distribution from a continuous one
- round up or down by 0.5
- etc. we can’t find exact probabilities like P(Y=6) when Y is continuous because the probability is effectively 0
- P(5.5
Approximating a Binomial using a Normal
To approximate, we just copy over the mean and variance of the Binomial to the Normal
If n is large and p is close to 0.5
X ~ B(n,p) –> Y ~ N(np = mean, np(1-p) = variance)
then you use normal distribution to find the probability like using the equation z = x - m / o
Approximating a Poisson using a Normal
To approximate, we use the same mean and variance for the Normal as the original poisson
If lambda is large
X ~ Po(lambda) –> Y ~ N(lambda = mean, lambda = variance)
then you use normal distribution to find the probability like using the equation z = x - m / o
