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Discrete versus Continuous
Discret: A set of countable values
ex: {1,2,3,…}
Continuous: Values in intervals
ex: (0,100)
Probability Function for a Continuous Random Variable and conditions
Probability Density Function (PDF) → not equal to Pr(X=x) since it can take on an infinite number of values
Denoted as f(x), fx(x)
What is a Discrete Uniform Distribution and its criteria?
1) Finite Number of possible values
2) Consecutively spaced by 1 (ex: 1,2,3,...,10 inclusive)
3) Equally likely to be observed
Cumulative Distribution Function (CDF) for a Discrete Random Variable
Denoted as P(X ≤ x), F(x), Fx(x)
Cumulative Distribution Function (CDF) for a Continuous Random Variable
Denoted as P(X ≤ x), F(x), Fx(x)
What are the two important formulas to know for a CDF for a discrete random variable?
How do we calculate the CDF probabilities for a continuous random variable?
What are the properties of a CDF for a Discrete Random Variable?
1) Non-decreasing
2) Fx(-∞) = 0, Fx(∞) = 1
What are the properties of a CDF for a Continuous Random Variable?
1) Non-decreasing
2) Fx(-∞) = 0, Fx(∞) = 1
What is the survival function for a discrete random variable?
What is the survival function for a continuous random variable?
Hazard Rate Function
Only for continuous random variables
Likely not tested on the exam
What are mixed distributions?
1) Multiple distinct, discrete distributions
1) Multiple distinct, continuous distributions
1) A combination of discrete and continuous distributions
What are the 2 ways to understand a mixed distribution?
1) Focus on the individual distribution separately
2) View the mixed distribution as a whole
Definition: Expected Value
Also called the mean, expectation, or first moment
Denoted as E[x]
What is the expected value for a discrete random variable?
What is the expected value for a continuous random variable?
What are the 3 key properties for an exoected value?
What do we do if there is a mixed distribution?
Use intuitive and reasoning
If piecewise, split sum/integral wherever fx(x) for g(x) changes
What is the conditional expectation for a discrete random variable?
Use the same general approach
Use conditional probabilities and conditional ranges
What is the conditional expectation for a continuous random variable?
Use the same general approach
Use conditional probabilities and conditional ranges
What is the conditional PDF?
Survival Function method for Discrete Random Variables
Alternative to Probability Function Method
Can only use if the range is non-negative
Survival Function method for Continuous Random Variables
kth and 1st Raw Moment
kth and 2nd central moment
Definition: Variance
Average squared deviation from the mean
Variance Equations for x and g(x)
Properties for variance
Coefficient of Variation
Measures variability of a random variable
Percentiles
πp = 100th Percentile
Pr(x ≤ πp) = p
πp is the percentile value such that the cumulative probability up to that value is p
Percentile for Discrete Random Variables
Percentile for Continuous Random Variables
What are the common percentiles and IQR?
What is the mode for discrete random variables?
Most likely values of a random variable
Value of x maximizes the PMF/PDF
The highest value for px(x) is the mode
What is the mode for continuous random variables?
Critical Point with largest value of fx(x)
Skewness: Equation
Measures a distributions symmetry
What is zero skewness, positive skewness and negative skewness?
Kurtosis: Equation
Measures the “peakedness” of a distribution
What do the different levels of kurtosis mean?
What is an example of a discrete uniform distribution?
1) Rolling a fair die
2) The number of customers that visit a store during a workday is uniform on {20,21,...,80}
What are the parameters of a discrete uniform distribution?
x ~ Discrete Uniform(a,b)
a and b are the parameters
What is the PMF of a discrete uniform distribution?
What is the Expected Value (mean) of a discrete uniform distribution?
What is the Variance of a discrete uniform distribution?
What is a Bernoulli distribution and its criteria?
1) 2 possible outcomes, 0 and 1
2) 0 and 1 can be any binomial event
What is an example of a Bernoulli distribution?
A coin flip would be Bernoulli if we assign 1 to heads and 0 to tails and vice versa
What if the PMF of a Bernoulli distribution?
What if the mean of a Bernoulli distribution?
What is the Variance of a bernoulli distribution?
What is the Bernoulli Shortcut?
If we have an event with only 2 outcomes (a and b) then we can use this
What are the parameters of a Bernoulli Distribution?
x ~ Bernoulli (p)
Parameter is p
What are the parameters of a Binomial Distribution?
x ~ binomial(n,p)
n and p are the parameters
What is a Binomial Distribution and its criteria?
- The sum of n independent Bernoulli Trials with the probability of success p
- Counts the number of successes
What is an example of a Binomial Distribution?
Tossing a coin 5 times and counting the number of heads in those 5 trials
How can you identify a binomial distribution?
1) Fixed number of independent trials
2) Same 2 possible outcomes for each trial
3) Number of sccesses (varies) is the random variable
How can you identify independent trials? (2 ways)
1) sampling with replacement
2) Sampling without replacement from a large population of unknown size (while population changes, the impact on each trial is small - treat the population as staying the same)
What would dependent trials look like?
Sampling without replacement of a known population size
What is the PMF for a binomial distribution?
What is the expected value for a binomial distribution?
What is the variance for a binomial distribution?
What is the special property for the sum of a binomials?
What is a Hypergeometric Distribution and its criteria?
1) Same situation as binomial but with DEPENDENT Bernoulli Trials
2) Sampling WITHOUT replacement from a KNOWN population size
What are the parameters for a hypergeometric distribution and what do they mean?
x ~ Hypergeometric (N, m, n)
N is the population size
m is the successes in a population
n is the number of dependent trials
What is the PMF of a hypergeometric distribution?
What is the Expectation of a hypergeometric Distribution?
What is the Variance of a hypergeometric Distribution?
What other technique can we use to solve hypergeometric questions?
Counting techniques!
What is a Geometric distribution and its criteria?
The number of independent bernoulli trials when we want to get the FIRST success
1) Fixed number of successes
2) Variable number of trials
What is an example of a geometric distribution?
The number of coin tosses it takes to get the first “heads”
How can we identify a geometric distribution?
1) Indepenedent bernoulli trials
2) Same two possible outcomes for each trials
3) Random variable is the number of trials/failues to get to ONE SUCCESS
2) Number of trials or failures? (“Number of trials before the first success” → number of failures)
What is the parameter for a geometric distribution?
x ~ geometric(p)
p: probability of a success
What is the trial version PMF of a geometric distribution?
X = Number of Trials to get one Success
What is the trial version Expectation of a geometric distribution?
What is the trial version variance of a geometric distribution?
What is the Failure version PMF of a geometric distribution?
Y = Number of Failures before one success
What is the Failure version Expectation of a geometric distribution?
What is the failure version Variance for a geometric distribution?
What is the memoryless property for the trial version of a geometric distribution?
The outcomes of previous trials do NOT affect the probabilities of future trials
Consider every new trial as the first trial
What is the memoryless property for the failure version of a geometric distribution?
The outcomes of previous trials do NOT affect the probabilities of future trials
Consider every new trial as the first trial
What are the is the definition and parameters for a negative binomial distribution?
x ~ Negative Binomial(r,p)
The number of trials to get r successes, where each trial has a success probability of p
What is the failure version expectation of a negative binomial distribution?
What is an exmaple of a negative binomial distribution?
The number of coin tosses required to ger 5 “heads”
What is the trial version PMF of a negative binomial distribution?
What is the trial version expectation of a negative binomial distribution?
What is the trial version variance of a negative binomial distribution?
What is the special property for negative binomials?
What is the failure version PMF of a negative binomial distribution?
What is the failure version Variance of a negative binomial distribution?
How can you identify a negative binomial distribution?
1) Independent bernoulli trials
2) Random variable is the number of trials/failures to get a certain number of successes
Fixed number of successes, r
Variable number of trials/variables
What is a poisson distribution?
Number of events in a fixed interval
Occurences in disjoint intervals are independent
Interval is usually time but it can be something else
What is an example of a poisson distribution?
The number of tornados in a field during a year
Count the number of flowers in a square foot of a field
What is the parameter of a poisson distribution?
x ~ poisson(λ)
λ (lambda) is the parameter, it is the mean (event rate)
What is the PMF of a poisson distribution?
Typically told to use poisson
Recognize from PMF
What is the expectation of a poisson distribution?
What is the variance of a poisson distribution?
What are the 2 properties of a poisson distribution?
What is a continuous uniform distribution?
DO NOT CONFUSE WITH DISCRETE DISTRIBUTIONS
Equal-length intervals are equally likely
Infinite number of possible values (any value between a and b)
say the time to drive to work is between 8 and 12 minutes
Probability between 8 and 9 minutes, 9 and 10 minutes, 10 and 11 minutes, and 11 and 12 minutes are all the same
What are the parameters of a and b?
x ~ continuous uniform(a,b)
a is the smallest value, b is the largest value
What is the PDF of a continuous uniform distribution?
Note 
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