STAT MOD 4 Flashcards
What are the two types of random variables?
1) Discrete
2) Continuous
Assigns a number to each outcome of a random circumstance or to each unit in a population
Random variable
Something that we collected about our units that varies from unit to unit
- describe probabilities to outcomes of that random variable
variable
What letter do you use to refer to the random variable?
What letter do you use to refer to a particular value of the variable?
1) X (capital) for random variable itself – abstract sense (height)
2) x (lowercase) for particular value of that variable — actual height
Type of random variable that has possible values that are ISOLATED POINTS on a number line
- can be counted
- find probabilities for exact outcomes
discrete random variable
Type of random variable where possible values are all points in some interval
- limited to finding probabilities for intervals
- probabilities for exact outcomes are zero
continuous random variable
Notations for discrete random variables:
What is X? What is k? What is P(X=k)?
X = random variable
k = a number the discrete RV could assume
P(X = k) is the probability that X equals k
What is probability distribution function (pdf)?
a table that assigns probabilities to possible values of X
k = list of possible values, outcomes
P(x = k) the whole column has to equal 1
What are pdf requirements?
1) sum of the probabilities over all possible values of a discrete RV = 1
2) probability of any specific outcome for discrete RV must be between 0 and 1 – no negative probability
What are the steps to a sample space?
1) List all simple events in sample space
2) Identify the value of the random variable X for each simple event
3) Find the probability for each simple event
4) Find P(X = k) as the sum of the probabilities for all simple events where X = k
What is cumulative distribution function (CDF)?
Table that gives P(X ≤ k) for any real number k
- highest value for CDF should always be 1
Can you make a list of k for continuous RV?
Cannot make a list of k (not using pdf) because the outcome can be any value in an interval
How do you represent the probability of the RV taking values in an interval?
Use the area under a density curve
What is a density curve?
a curve that specifies a probability distribution for a continuous RV
- f(x) ≥ 0
- total area under the density curve = 1
What is the probability that x falls in any particular interval?
probability that x falls in any particular interval is the area under the density curve in that interval
How is the mean value of a random variable denoted? What else is it called?
- µ denotes mean value of RV
which describes where the probability distribution of x is centered
- also called expected valuable of the RV – E(X)
How is the standard deviation of a RV denoted? What does it describe?
- 𝜎 denotes SD of RV
describes the variability in the probability distribution
How do you find the expected value?
create probability distribution function table
then multiply across: kP(X = k) then add it all up to get expected value
What is the standard deviation of a random variable?
Roughly the average distance the random variable falls from its mean/expected value over the long run
How do you find standard deviation?
StDev(X) = 𝜎 = √(k - µ)^2 * p(x=k)
What is a normal random variable? What kind of distribution does it have?
type of continuous random variable that has a bell-shaped probability density curve (called a normal curve)
normal RV has a normal distribution
What are characteristics of normal distribution?
- symmetric/bell-shaped
- unimodal
characterized by its mean and standard deviation
What is the probability of x for normal random variables?
P(X ≤ µ) = P(X ≥ µ) = 1/2
What is the empirical rule? Draw it
68-95-99.7 rule
probability of falling within any particular number of standard deviations of µ is the same for all normal distributions
