Math 356 GR2

List and explain in your own words the two probability axioms:

1. 𝑃(𝐴)≥ 0 each event must have a probability greater than or equal to zero 2. 𝑃(𝑆)=1 the entire sample space has a probability summing up to 1 (100%)

A and B are independent events IF AND ONLY IF (write two equations that this is true for)

1. 𝑃(𝐴|𝐵)=𝑃(𝐴) 2. 𝑃(𝐴∩𝐵)=𝑃(𝐴)∗𝑃(𝐵)

Define independent events

Events are independent if the probability of one event occurring does not change whether or not the other event has occurred

Describe the mean of a random variable in your own words

The mean is the long-run average value of several rolls of a random variable

Describe how the cdf can be used to generate values of a discrete random variable

The range of a cdf is a subset of [0,1], so generate a random number between 0 and 1 and apply an inverse function to the random variable's support The interval [0,1] is divided into subintervals of width corresponding to the probability of each outcome. The probability uniformly distributed random number lands into one of these subintervals is thus the same as the probability of the corresponding outcome

Describe the variance of a random variable in your own words

The variance of a random variable is a number that describes its variability about the mean. Random variables with more variance have a larger spread.

Let X be a random variable given by the pdf: 𝑓(𝑥)={𝑒−𝑥, 𝑥≥0 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 Compute E(X) Compute E(X2) Compute Var(X)

𝑬(𝑿) = ∫𝑥𝑒−𝑥𝑑𝑥=∞01 𝑬(𝑿𝟐) = ∫𝑥2𝑒−𝑥𝑑𝑥=∞02 𝑽𝒂𝒓(𝑿)=𝑬(𝑿𝟐)−𝑬(𝑿)𝟐= 2−1= 1

What is the formula for E(g(X))? What does this mean?

𝑬(𝒈(𝑿))=∫𝒈(𝒙)𝒇(𝒙)𝒅𝒙 𝑺 where f(x) is the pdf of X. This integral computes the average of g(X) weighted by the pdf of X.

List the 6 Probability properties in your own words

1. Probability of an empty set is zero 2. The probability of the complement of event A is 1 minus the probability of event A 3. If event A is a subset of event B, then the probability of event A is less than or equal to the probability of event B 4. The probability of the union of event A and event B is equal to the probability of event A occurring, plus the probability of event B occurring minus the probability of both occurring (subtract the probability that was double counted) 5. The probability of an event A is the sum of the intersections of A with the events B and event B's complement (𝐵𝐶) that fills up the sample space. 6. The probability of the complement of the union of two events is the probability of the intersection of the complements of those two events. Also, the probability of the complement of the intersection of two events is equivalent to the probability of the union of the complements of those events.

What is the formula for expectation?

E(X) = ∑𝒙𝒇(𝒙)𝒙 The sum of the product of x and the probability of x occurring over all events x in the support

Let X be the number of heads in three coin flips. Compute E(X2):

E(X2) = 0*1/8 + 1*3/8 + 4*3/8 + 9*1/8 = 3

What is the linear transformation formula for expectation?

E(aX+b) = aE(X)+b In words: 1. Scaling all support elements by 'a' scales the mean by the same factor. 2. Shifting all support elements by 'b' shifts the mean by the same amount.

What is the outcome and what is the event in terms of the sample space?

Each possible result is an outcome. An event is a set of outcomes

What are moments?

Moments are a single number used to describe a population (i.e. a random variable).

What is the difference between Probability and Statistics?

Probability deals with predicting the likelihood of future events (aka, probability uses known population information to explain what we expect to occur). Statistics involves the analysis of the frequency of past events (aka, statistics uses known sample data and tries to describe what is happening.

Let X be the number of heads in three coin flips. Compute Var(X):

Recall µ = E(X) = 1.5 Var(X) = E[(X- µ)2] = (0-1.5)2*1/8+(1-1.5)2*3/8 + (2-1.5)2*3/8 + (3-1.5)2*1/8 = 3/4

What is a cumulative distribution function?

The cumulative distribution function (cdf) of any random variable X (discrete, continuous, or other) is a function FX(x) = P(X ≤ x). In words, the cdf is the function that returns the probability that a random variable is less than or equal to the input. The cdf is a right-continuous, non-decreasing function that can take on values between 0 and 1 inclusive.

Provide an example of a continuous random variable

The height of a person, the amount of time it takes to drive to work, and the lifespan of a lightbulb can each be thought of as continuous random variables

Define the following terms: ⋃, ∩, ⊂, ∅ ={}, 𝐴𝑐 = 𝐴′

⋃ = union ∩ =𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 ⊂ = subset ∅ ={}= 𝑒𝑚𝑝𝑡𝑦 𝑠𝑒𝑡 𝐴𝑐 = 𝐴′= complement

Another representation of Bayes Formula

𝑃(𝐷|𝑇)= 𝑃(𝑇|𝐷)∗𝑃(𝐷)/(𝑃(𝑇))

What is Bayes Formula as provided in the text?

𝑃(𝐷|𝑇)= 𝑃(𝑇|𝐷)∗𝑃(𝐷)/(𝑃(𝑇|𝐷)∗𝑃(𝐷)+𝑃(𝑇|𝐷′)∗𝑃(𝐷′))

What is the equation for permutation? (define the variables)

𝑛𝑃𝜅=𝑛!(𝑛−𝑘)! Where n is the group size and k is the number of objects selected

What is another word for "domain"?

Support

what does "x" mean?

any outcome in that support

What is the equation for combination? (define the variables)

(𝑛𝑘)= 𝑛!(𝑛−𝑘)!𝑘! Where n is the group size and k is the number of objects selected

What is the formula for conditional probability? Be sure to define the notation

-where P(A|B) reads probability of A given event B occurred - 𝑃(𝐴∩𝐵) is the probability of event A and event B occurring (Joint probability) also called the intersection of events

What are the three main properties of the pdf? Give the mathematical definitions and describe them in your own words

1. f(x) ≥ 0. A pdf must be non-negative because negative densities don't make sense. 2. ∫𝑓(𝑥)𝑑𝑥 𝑆=1. Integrating the pdf over its supports accounts for all possible outcomes. 3. 𝑃(𝑋∊𝐴)=∫𝑓(𝑥)𝑑𝑥 𝐴 which implies 𝑃(𝑎≤𝑋≤𝑏)=∫𝑓(𝑥)𝑑𝑥𝑏𝑎. The probability of an event is the area underneath the pdf curve.

What is a random variable?

A random variable X is a function X:S→R that assigns exactly one number to each outcome in an experiment

Define and give an example of a sample space

A sample space is the set of all possible results from an experiment. If we are rolling a die the sample space would be 𝑆={1,2,3,4,5,6}

What is the domain/support for a continuous random variable?

All real numbers in an interval

A|B is or is not the same as B|A?

A|B is NOT the same as B|A

How may continuous random variables be simulated?

Continuous random variables may be simulated by generating a random number on [0,1] and applying the inverse cdf.

What is the distinction between discrete and continuous random variables?

Discrete random variables have a countable support. The support of a continuous random variable is a continuous interval or group of continuous intervals

Let X be the number of heads in three coin flips. Compute E(X):

E(X) = 0*1/8 + 1*3/8 + 2*3/8 + 3*1/8 = 1.5

Define independent:

Events are independent if the outcome of the event is not impacted by previous event outcomes.

what is F(X) used for?

F(X) is used for cdf (cumulative distribution function)

How do moments differ from summary statistics?

Moments are based on the population while summary statistics are computed from a sample

Is 𝑓(𝑥)={sin(𝑥), 0≤ 𝑥≤𝜋0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 a valid pdf? If so, what is the cdf? If not, why?

No, the pdf presented is not valid. Although 𝑓(𝑥)≥0 , ∫sin(𝑥)𝑑𝑥=2≠1𝜋0, violating the second pdf property.

Let Y be the sum of a roll of three fair, six-sided dice. What does P(10 < Y < 15) mean

P(10 < Y < 15) is the probability that the sum of the dice is greater than 10, but less than 15 (i.e. 11, 12, 13, or 14).

Let X be the number of heads in three coin flips. What is P(X = 2)

P(X = 2) = fX(2) = 3/8 because there are three ways to flip exactly two heads (HHT, HTH, THH) and there are eight total ways to flip the three coins

Let X be the number of heads in three coin flips. What is P(X > 1)?

P(X > 1) = 1 - P(X ≤ 1) = 1 - FX(1) = 1 - 1/2 = 1/2

Provide an example of a discrete random variable

Rolling a die, flipping a coin, and the number of people who enter a store in an hour can each be thought of as discrete random variables

What is the cumulative distribution function (cdf) of a continuous random variable, X? How does it differ from the cdf of a discrete random variable?

The cdf is F(x) = P(X ≤ x). In words this is the probability that the random variable, X, is less than or equal to a particular value, x The definition is the same as the cdf used for discrete random variables

How is the cdf of a continuous random variable X defined mathematically?

The cdf is 𝑃(𝑋≤𝑥)=∫𝑓(𝑡)𝑑𝑡𝑥−∞=𝐹(𝑥). In practice, the lower bound of integration may be replaced with the least element of S when such an element exists.

Which two notations are used for the mean of a random variable X?

The mean is denoted as E(X) and as µ

What is a probability mass function?

The probability mass function (pmf) of a discrete random variable X is a function fX(x) = P(X = x). In words, the pmf is a function that assigns probability to each possible outcome of X

Let X be a continuous random variable. What is P(X = x)?

The probability that a continuous random variable equals a particular value is 0 because continuous random variables do not have point masses. In other words, the area underneath a point is 0. Mathematically, 𝑃(𝑋=𝑎)=𝑃(𝑎≤𝑋≤𝑎)=∫𝑓(𝑥)𝑑𝑥=0𝑎𝑎.

Which three notations are used for the variance of a random variable X?

The variance is denoted as σ2, Var(X), or E[(X- µ)2]

What is the linear transformation formula for variance?

Var(aX+b)=a2Var(X) In words: 1. Scaling all support elements by 'a' scales the variance by 'a2'. 2. Shifting all support elements by 'b' does not affect their spread.

If you see E(X), what does it mean?

WEIGHTED AVERAGE multiply probability by x and sum them up

Describe rule 1 in your own words:

When order matters but all options are available(replaced) you multiply the number of options available for each entry.

What is the R command for combination?

choose (n, k)

What command in R can be used to repeat something, as you want to do for a simulation?

do (number of times)

what is f(x) used for?

f(x) is used for pmf (probability mass function-single point mass so discrete) and pdf (probability density function)

How would you get R Studio to integrate x^2/18 from -3 to 1?

integrate(function(x)x^2/18,-3,1)

what does "X" mean?

random variable