Lecture 1: Fundamentals of Statistics: Probability and Distribution Flashcards
Events are independent when…
the occurence of one does not affect the probability of occurence of the other
How do you calculate the probability of two independent events occuring?
P(a)P(b)
Probability of a times probability of b
What is conditional probability? What is its formula?
Conditional probability is a measure of the probability of an event ocurring, given that another event has occured.
It is read as “Probability of A given B”
P(A|B) = P(A{intersect}B)/P(B)
Example: If A is Rain and B is cloudy, what is the probability of Rain given that its Cloudy?
What are permutations? What is its formula?
Permutations are an arrangement of r objects from a pool of n objects in a certain order
Formula:
P(n,r) = n!/(n-r)!
Example:
Passcode
What are combinations? What is its formula?
Combinations are an arrangement of r objects from a pool of n objects where order does not matter
Formula:
C(n,r)=P(n,r)/r!=n!/r!(n-r)!
Example: Seeing how many ways you can divide a class of students into groups of five
Describe Bayes’ theorem
Bayes’ theorem (Also called Bayes’ law/rule)
It provides a way to update an estimate of an event A occurring when new evidence restricts your sample space to B
In Bayesian statistics, probability is viewed as a degree of belief of an even occurring
What is Bayes’ Theorem formula?
P(A|B) = P(B|A)P(A)/P(B)
What is the law of total probability?
If event A and B are mutually exclusive and collectively exhaustive, then
P(A) = P(A|B)P(B)+P(A|B^c)P(B^c)
- Acould mean the event “Patient has liver disease.” Past data tells you that 10% of patients entering your clinic have liver disease. P(A) = 0.10.
- Bcould mean the litmus test that “Patient is an alcoholic.” Five percent of the clinic’s patients are alcoholics. P(B) = 0.05.
- You might also know that among those patients diagnosed with liver disease, 7% are alcoholics. This is yourB|A:the probability that a patient is alcoholic, given that they have liver disease, is 7%.
What is A|B?
P(disease|alcoholic) = P(0,07)P(0,1)/P(0,0,5) = 0,14
What is a random variable?
What is a probability distribution?
random variable = is a mapping from sample space S to real number (e.g., it can represent all the possible outcomes of a random experiment)
probability distribution = the mathematical function that gives the probabilities of occurrence of different outcomes of an experiment
Name a couple of the different types of distribution
- Discrete Distributions **
- Discrete uniform distribution
- Bionomial Distribution
- Poisson Distribution
- Continuous Distributions **
- Uniform Distribution
- Gaussian Distribution (Normal Distribution)
- Exponential Distribution
What does the central limit theorem say?
“if you take sufficiently large random samples from a population, then the distribution of the sample means will be approximately normally distributed even if the population does not follow a normal distribution”
What is the Probability density function?
the probability that a random variable X receives a certain value
What is the Cumulative distribution function?
the probability that a random variable X receives a value less or equal to a certain value
When are two events collectively exhaustive?
A set of events is collectively exhaustive if they cover the whole probability space (when their union equals the sample space)
e.g., when you roll a die, the set of events 1, 2, 3, 4, 5, 6 is collectively exhaustive as it encompasses the entire range of possible outcomes
