Unit 6 Stat

Discrete Random Variables:

  • Random Variable - variable that takes numerical values determined by the outcome of a chance process.

  • Discrete Random variable - has a countable number of possible values

  • Continuous Random Variable - has an uncountable (infinite) number of possible values. Normally capitalized, eg. X, Y, Z

Probability Distribution of a Random Variable:

  • distribution - the set of all possible values of that variable and how often it takes them. Lists the values and their probabilities.

  • the probabilities must be a number between 0-1 (inclusive) and all of the probabilities must sum to 1

Expected Value (Mean) of Discrete Random Variables:

  • the expected value is the mean/avg of long term repeated outcomes

Variance of Discrete Variables:

  • variance =

Linear Transformations:

  • linear transformation - an operation of the form ax+b on a variable (x), where a and b are constants.]

Linear Transformations of Random Variables:

measures of spread (standard deviation, range, IQR) - are affected by multiplication and division

measures of center (mean, median, mode) - are affected by addition, subtraction, multiplication, and division

Binomial Setting: scenarios that can be broken down into two outcomes: success and failure

Binomial Experiment - a probability experiment that satisfies all four of the below requirements

  • B - Binary: each trial can only have two outcomes, success or failure

  • I - Independent: the outcomes of each trial must not affect other outcomes, check 10% Condition if sampling without replacement.

  • N - Number: number of trials must be fixed

  • S - Same: the probability of success must remain the same for each trial

Notation:

P(S) - probability of success

P(F) - probability of failure

p - numerical probability of success

q - numerical probability of failure

P(S) = p

P(F) = 1 - p = q

n = number of trials

X = number of successes

n! = factorial notation

Binomial Coefficient:

  • a factorial is an operation where you multiply positive whole numbers in descending order down to 1

    • ex. 4! = 4 × 3 × 2 × 1

  • combinations are an arrangement of objects in which the order doesn’t matter

Mean and Standard Deviation of Random Binomial Variables:

  • check with BINS that it is a binomial variable

  • mean is the # of successes we expect out of n trials

    • μx= np

  • SD is how much we vary from the mean on average

    • σx

Normal Approximation of Binomial Distribution:

Relative Independence -

  • Independence Criterion (10% Rule) - sampling without replacement

    • if the sample is less than or equal to 10% of the population. Used when checking the I in BINS for a binomial distribution.

Large Counts Condition - both the expected successes and failures are at least 10. This condition allows us to use a normal distribution to approximate the binomial distribution.

  • when n (# of trials) is large, the distribution of X (# of successes) is approximately normal with 𝜇X = np, 𝞂X = √(np(1 - p))

  • the normal distribution can be used to approximate the binomial distribution when np ≥ 10 and n(1 - p) ≥ 10

Geometric Random Variables: a geometric probability model (and a geometric random variable) must satisfy all four of these requirements -

  • each trial can only have two possible outcomes - success and failure

  • the outcomes of each trial must be independent of each other

  • the trials are repeated until a single success is achieved

  • the probability of success must remain the same for each trial

Mean and SD of Geometric Random Variables: If X has a geometric distribution with a parameter (p) then, P(X = x) = P(1-p)x-1p where x = 1, 2, 3…

  • Mean - μx = 1/p

  • SD -