Probability Flashcards
What is Continuous Probability Distribution?
- probability distribution in which random variable X takes on any value + describes it {P(X = x) = 0}
- uses probability density function
What is the discrete probability distrubution?
probability distribution counts occurrences that have countable or finite outcomes
What are the features of a continuous distribution?
- normal distribution
- Uni-modal & symmetrical
- extreme values away from centre
What is the standard normal distribution for continuous distribution probability?
σ = 1, μ = 0
What happens when the mean is changed on a normal distribution?
The curve changes where it is centred on the x-axis
What happens when the standard deviation is changed on a normal distribution?
Adjusting σ changes the shape of the curve
List the properties of a normal distribution
- 68% -> 1 SD either side of mean
- 95% -> 2 SD either side of mean (1.96 exactly)
- 99.75% -> under 3 SD but extremely rare
What are the 3 properties of sampling distribution?
- variability in sample estimates
- higher sample size = less variable in samples -> more representative sample
- larger samples = point-estimate from samples are closer to population value
What is the difference between a smaller and larger standard error?
Large standard error = more distributed
Small standard error = grouped in one area; it’s closer to the population parameter
What happens when the number of samples increase?
- sampling distribution becomes normal
- samples x̄s pile around µ
- the SE of sampling distribution becomes narrower
What is the central limit theorem?
Taking large samples from a population, the sample mean will be normally distributed even if the population isn’t
What is analytic probability?
- Probability of event equal to ratio of successful outcomes to all possible outcomes
What is the law of large numbers?
- larger the number the more proportion represents ground truth, representing the world
What is the range probability rule and what does it mean?
- probability of any event falls between 0 and 1
- P(A) approaches 1 means more likely to occur
What is the Sum of outcomes probability rule and what does it mean?
- Sum of probabilities of all possible outcomes = 1
- all sample includes all possible outcomes of experiment so some outcomes must occur
What is the complement rule probability rule and what does it mean?
- Probability of A^c(NOT A) =1- probability of A
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What is the Simple addition rule probability rule and what does it mean?
- Probability one or both events occur
- adding probability of one event to another
- used for mutually exclusive events as can’t coexist together
What is the general addition rule probability rule and what does it mean?
- Probability one or both events occur when events not mutually exclusive
- Also used when events are mutually exclusive -> probability both occur together is 0
What is the multiplication rule probability rule and what does it mean?
- Probability of intersection of A and B if they are independent events.
- Probability of A and B always less than or equal to the probability of either single events
What is the conditional probability rule and what does it mean?
- The probability of one of the other happens
- Multiply probability of A with the probability of B given A
What is the conditioned probability?
Probability of B given A
- events are dependent on likelihood of one event changing based on the outcome of the other event
What is the difference between the conditioned probability and multiplication rule?
- When A and B are independent P(B|A) = P(B) is the simple multiplication rule BUT
conditioned probability when A and B are dependent
What is Bayes’ rule?
- Calculating conditional probabilities when intersection unknown
- Follow on from conditioned probability
- Allows update assessment of probability as more evidence gathered
What is a sample space?
All possible outcomes
What is a simple event?
singular event
What are discrete distributions?
- mapping values of random variables to probability of it occurring
- range of p across variables
What is the probability mass function?
- probability each discrete random variable exactly equals specific value
What does this mean f(x)= P(X= x)
X = random variable
x = specific event in X
What is the process of the probability mass function?
- Sum frequencies
- Divide frequency of each outcome by total frequency
- Provided with discrete probability distribution
What are the characterisitcs of binomial distributions?
- two outcomes: success or failure
- fixed number of observations
- observations are independent
- probability of success same as each observation
What is the interest in binomial distributions?
number of successes given a fixed number of trials
What is the cumulative probability function?
Total probability of all values before or after a given point
Sums the probability of individual outcomes
What is the difference between continuous probability distribution vs. discrete probability distribution?
Discrete: describes random variables producing discrete set of outcomes vs. continuous describes random variable producing continuous set of outcomes.
What is used to describe continuous distributions?
Probability density function
What is used to describe discrete density functions?
- Probability mass function
- Binomial distribution
What change occurs when the mean is changed on a normal distribution?
changing the mean changes where the curve is centred on x-axis
What change occurs when the standard deviation is changed on a normal distribution?
changes the shape of the curve
- Higher SD = flatter curve
- Lower SD = higher curve
What do continuous random variables tell us?
Probability of a range of scores occuring
What is needed for the value under the curve for continuous distribution?
the integral and probability density function
Why are z-scores used?
- present normal distribution
- standardise the value of x
What is a point-estimate?
- sample value of variable of interest
What is a population parameter?
- population value of variable of interest
What is a sample?
- subset of population to collect data to make inference
What is a sampling distribution?
probability distribution of stats obtained from sampling population repeatedly
