Exam 1 Flashcards
get an A?
What notation is used to denote an open interval?
(a, b) means the interval of all numbers x where a < x < b
What notation is used to denote a closed interval?
[a, b] means the interval of all numbers x where a ≤ x ≤ b
What is the definition of a set?
A set is a collection of elements.
What symbols are used to denote a finite set?
Curly braces { } are used to denote a finite set.
What is the notation for the empty set?
The empty set is denoted by either { } or ∅.
True or False: The empty set is a subset of any nonzero set.
True
If A = {0, 1}, how many total subsets does A have?
Four total subsets.
What is a statistical experiment?
A process which generates a single outcome where there is more than one possible outcome, the possible outcomes are known in advance, and the outcome cannot be predicted with 100% certainty.
What is the sample space in a statistical experiment?
The set of all possible outcomes of the experiment.
What does an event represent in statistics?
An event is any subset of the sample space of a statistical experiment.
What does it mean for two events A and B to be contained in each other?
Event A is contained in event B if every outcome in A is also in B.
What is the complement of an event A?
The set of all outcomes in the sample space that are NOT in A.
What is the union of two events A and B?
The set of all outcomes in either A or B or in both.
What is the intersection of two events A and B?
The set of all outcomes in both A and B.
What does it mean for events A and B to be mutually exclusive?
Events A and B cannot both occur simultaneously, meaning their intersection is empty.
What are DeMorgan’s Laws?
(A ∪ B)’ = A’ ∩ B’ and (A ∩ B)’ = A’ ∪ B’
Fill in the blank: A set is said to be a ______ of a set if all the elements in A are also contained in the set B.
subset
What does the notation ∅ represent?
The empty set.
What happens to the sample space when the outcomes are infinite?
The sample space may contain an infinite number of outcomes.
What are the three ways to define probability?
- Probability as a long-term relative frequency
- Subjective probability
- Axiomatic approach to probability
Define probability as a long-term relative frequency.
The proportion of times an event occurs in a large number of trials.
What is subjective probability?
A type of probability based on personal judgment or experience.
How can personal probabilities change?
They can be updated as more information is gathered.
What is Bayesian statistics?
A framework for updating probabilities in the presence of new information.
What are the fundamental axioms of probability?
- P(A) ≥ 0
- P(S) = 1
- If A and B are mutually exclusive, P(A ∪ B) = P(A) + P(B)
What is the probability of the empty set ∅?
P(∅) = 0.
In a finite sample space, how is the probability of an event defined?
P(A) = # of outcomes in A / # of total outcomes in S.
What is the formula for the probability of an intersection of two events?
P(A ∩ B) = # of outcomes in A ∩ B / # of outcomes in S.
In a finite sample space with equally likely outcomes, how is the probability of an event defined?
P(event) = # of outcomes in event / # of total outcomes in sample space.
What is the Fundamental Counting Principle?
If there are n tasks with k1, k2, …, kn ways to complete each task, then the total number of ways to complete all tasks is k1 × k2 × … × kn.
What is a permutation?
The arrangement of distinct items in a specific order.
What is the formula for finding the number of permutations of k items from a set of n items?
P(n, k) = n! / (n - k)!
What distinguishes combinations from permutations?
Order matters in permutations but not in combinations.
What is the formula for combinations?
C(n, k) = n! / (k! (n - k)!)
Define a partition in the context of combinatorial counting.
Ways to place distinct items into groups without regard to order.
What is the formula for counting partitions?
P = n! / (n1! n2! … nk!)
What is the general approach to calculate probabilities when the sample space is finite?
P(event) = # of outcomes in event / # of total outcomes in sample space.
What is conditional probability
the probability of an event occuring given that another event has already occurred.
P(A∣B)= (P(A∩B)) / (P(B))
What does it mean for two events to be independent
when the occurance of one does not affect the probability of the other
P(A∩B)=P(A)×P(B)
P(A∣B)=P(A)
P(B∣A)=P(B)
What is the difference between independence and mutual exclusivity
Independence means the occurrence of one event does not affect the probability of the other, while mutual exclusivity means the two events cannot occur together (i.e., P(A∩B)=0
P(A∩B)=0). Independent events can occur together, but mutually exclusive events cannot.
What is conditional independence?
Two events A and B are conditionally independent given event C if:
P(A∩B∣C)=P(A∣C)×P(B∣C)
P(A∩B∣C)=P(A∣C)×P(B∣C)
Conditional independence does not imply marginal independence.
What is Bayes’ Theorem?
Bayes’ Theorem relates the conditional and marginal probabilities of two events. For events A and B, it is given by:
P(A∣B)=(P(B∣A)×P(A)) /(P(B))
where
P(B)=P(B∣A)×P(A)+P(B∣A^c)×P(A^c)
What is a random variable?
A numeric encoding of an experiment’s outcome, mapping sample space to real numbers
What is the support of a random variable
the set of all possible values it can take
What is the difference between a discrete and continuous random variable
Discrete: support is a list of numbers ( finite or countably infinite)
continuous: Support is an interval or union of intervals
What is a probability distribution for a discrete random variable?
Specifies values and their probabilities, with P(x)∈[0,1] and ∑P(x i)=1.
What is a Bernoulli trial?
A statistical experiment with two possible outcomes: “success” or “failure.” The probability of success is p.
What is the support of a Bernoulli random variable?
The support is {0,1}, where:
1 = success,
0 = failure.
What is the PMF of a Bernoulli random variable?
P(X=x)=p^(x)*(1−p)^(1−x) for x∈{0,1}.
What are the expected value and variance of a Bernoulli random variable?
E(X)=p,
Var(X)=p(1−p)
What is a Binomial random variable?
A Binomial random variable X counts the number of successes in n independent Bernoulli trials, each with success probability p. It is denoted as X∼Binomial(n,p)
What is the support of a Binomial random variable?
The support is {0,1,…,n}, where n is the number of trials.
What is a Poisson random variable?
A Poisson random variable X counts the number of occurrences of an event in a fixed interval of time or space, where events occur independently at a constant mean rate λ. It is denoted as X∼Poisson(λ).
What is the support of a Poisson random variable?
The support is {0,1,2,…}, which is countably infinite.
What is the scaling property of a Poisson random variable?
If X∼Poisson(λ) counts events per unit time/space, then Y∼Poisson(tλ) counts events over t units of time/space.
What is a continuous random variable?
A random variable whose support (set of possible values) is an interval or union of intervals. It cannot list all possible values, unlike a discrete random variable.
What is the probability density function (PDF) of a continuous random variable?
A function f(x) that describes the probability distribution of a continuous random variable. It satisfies: f(x)≥0 for all x,
Area under curve =1
What is the probability P(a≤X≤b) for a continuous random variable?
It is the area under the PDF curve between a and b
What is the probability P(X=x) for a continuous random variable?
P(X=x)=0 for any specific value
x, because the probability is spread over an interval.
What is the cumulative distribution function (CDF) of a continuous random variable?
The area under the curve up to that point
What is the probability P(X>x) for an exponential random variable X∼Exponential(λ)?
P(X>x)=e^(−λx)
What is the support of a uniform random variable X∼Unif(a,b)
The support is the interval [a,b]
What is the support of an exponential random variable X∼Exponential(λ)?
The support is [0,∞).
