S1 DAPR1 lecture 8 : discrete probability distributions

in stats, what does an experiment refer to

any process whose outcome is not known in advance, it has 3 features

what are the 3 features of an experiment in stats

1. there is more than one possible outcome (otherwise outcome is not unknown); 2. all possible outcomes are specified in advance; 3. each outcome occurs with some probability (p)

random experiments

the process of sampling simple events from a sample space, thus producing an outcome

random variable

a set of values that quantify the outcome of the random experiment - allows you to amp the outcome of a random experiment to numbers - e.g. 0 and 1 - characters is a capital letter

a random variable can be either (with examples)

discrete (e.g. coin toss, eye colour, no of children) or continuous (measuring heights with an ‘infinitely precise’ ruler)

probability distributions

map the values of a random variable to the probability of it occurring

what does a probability distribution do for discrete values

it maps a particular probability to a specific value of outcome via a probability mass function

the process and function is slightly different for

continuous distributions

a probability distribution maps the values of a random variable to

the probability of it occurring, based on the probability mass function

what is the pribability mass function

F(x) = P(X=x)

what does the probability mass function tell us

tells us the probability of the random experiment resulting in x

examples of 2 probability rules that probability functions follow

the sum of the probability of all possible values is 1; for any subset A of sample space: the probability of subset A is the sum of all the probabilities of all the simple events x within A

notation for the 1st rule

sum of N where i=1 (f(xi)) = 1

notation for second rule

P(A) - sum of where i is an element of A (f(xi))

what are the properties of binomial/Bernoulli experiments of processes

2 outcomes (success and failure) where probability of success is (P); interested in the number of successes (k) given a number of fixed trials (n); uses the binomial equation

for binomials what do p,k and n stand for

p: probability of success; k: number of successes; given a number of n fixed trials

E.g. of use of binomial

how many heads in series of coin tosses

binomial equation

f(k, n, p) = PR(X= k) = vector(n, k) p^k x q^(n-k)

what does q represent in binomial

1-p - probability if failure

vector n, k meaning for binomial

n choose k - the number of ways to select k successes from n observations

e.g. if n=5 and k=3, how do we work out the possible outcomes

using factorials, could be trials 1,2,3 correct or 2,3,4, correct etc

what odes n! mean where n=5

n factorial : 5 x 4 x 3 x 2 x 1

equation to find n choose k

n!/(k!(n-k)!

what do we do after finding n choose k

then plug this into the equation and calculate probability of getting 3/5 trial correct