STAT MOD 4: Chapter 6 Flashcards
What are random variables?
assigns a number to each outcome of a random circumstance to each unit in a population
What does a random variable describe?
Describe probabilities of outcomes of that variable
What is X? What is x?
X (refers to abstract random variable)
- height
x (refers to particular value of the variable)
- 7 foot
What is discrete random variable? Can we find probabilities for a discrete random variable?
possible values are isolated points on the number line; can take one of a countable list of distinct values
- can find probabilities for exact outcomes
What is continuous random variable? Can we find probabilities for a continuous random variable?
possible values are all points in some interval
- limited to finding probabilities for intervals of values
- probabilities of exact outcomes are zero
What is probability distribution function (pdf)? Draw probability distribution function
table that assigns probabilities to possible values of x
What do the following stand for in pdf?
- X
- k
- P(X=k)
- X = random variable
- k = a number the discrete random variable could assume
- P(X = k) = the probability that X equals k
What is cumulative distribution function (cdf)?
a table that gives P(X ≤ k) for any real number k
What are requirements that a discrete distribution (pdf) must satisfy?
All probabilities between 0 and 1
- impossible to have a negative outcome
Probability of all possible outcomes sums to 1
- All the values on P(X = k) side of the table must add up to 1
How do you compute mean/expected value of discrete random variable using a probability distribution table?
expected value: describes where the probability of distribution of x is centered
- Make pdf table of k and P(X = k)
then multiply each possible value (k) by its probability (P(X = k) → kP(X = k) - Add all multiplied values together to get mean/expected value of a discrete random variable
How is a probability of a continuous random variable expressed as?
Density curve: a curve whose area represents the probability of the continuous random variable taking values in that interval
What are characteristics of a density curve?
- The curve always falls at or above zero (can never be negative)
- The total area under density curve is equal to 1
- The probability a random variable (x) falls in a given interval is equal to the area under the density curve in that interval
What is uniform distribution? How do you find probability?
a square or rectangular density curve (ex: trolley waiting time)
Know the height and length of the shaded to find probability
How do you find areas under a given normal distribution (left tail, interval, or right tail)?
- normal problems
Use empirical rule - (if values perfectly fall on standard deviation values)
normalcdf (lower, upper, mean, SD)
- Use normalcdf when trying to find an interval
- use normalpdf when trying to find probability at an exact value
How do you find quartiles of any normal distribution?
How to find x value from the distribution given left or right area (inverse normal problems)?
- given percent as the left or right tail area
- find mean and standard deviation
- use invnorm(left tail area, mean, standard deviation)
- if given something lowest percent = left tail area, input value as is
- if given something highest percent = right tail area, subtract this from 1
How do you turn values from any normal distribution into z-scores from the standard normal?
Z-scores or standardizing: number of standard deviations that it falls from the mean
(x - μ/sigma) to get z-score then refer to standard normal distribution table for area of that z-score
How do you check for normality in a data set?
- Plot a histogram
- See if empirical rule fits the data
- Interpret a normal probability plot
- you are looking for the dots to fall on a 45 degree line (must follow a linear pattern) i
f the data set matches up to what you would expect a data set of that size to look like coming from a normal distribution.
What is binomial random variable?
defined as X = number of successes in the n trials of a binomial experiment
- x ~ bin (n,p)
What are requirements for a binomial experiment?
- Only two possible outcomes on each trial (success/failure)
- Fixed number of trials (n)
- Outcomes are independent from one trial to the next
- Probability of a success (denoted by p) remains the same from one trial to the next
- Probability of a failure is 1 - p
What are the commands to find to find binomial probabilities and cumulative probabilities?
○ Binompdf(n, p, k)
○ Binomcdf(n, p, k)
How do you compute mean of binomial distribution?
Mean (μ) = (trials)(probability success) = np
How do you compute standard deviation of binomial distribution?
Standard deviation (sigma) = square root of np (1-p)
Which conditions can the normal distribution be used to approximate binomial distribution?
If X has a binomial distribution and n is large, we can approximate the binomial probabilities
- If a SRS of size (n) is drawn from a large population and (n) is sufficiently large, the sampling distribution of X is approximately normal
○ Conditions:
- x ~ AN (np, square root of np(1–p)
if np ≥ 10 and np(1–p) ≥ 10
How do you use normal distribution to approximate the binomial distribution when appropriate?
Find approximate probabilities using the normal curve
First approximate the binomial distribution to normal distribution (check if np and np(1–p) ≥ 10)
Use either normalpdf or normalcdf (lower bound, upper bound, np, standard deviation or square root of np(1–p))
