Random Variables
Perfect positive linear dependency is indicated by what correlation?
+1.0
What is the range of correlation?
-1 to 1
Perfect negative linear dependency is indicate by what correlation?
-1.0
What does P(x) equal?
P(X=x) for all values of x
How do you represent conditional probability distribution of Y given that X=x?
P(y|x)=P(x,y)/P(x)
The expectation of Y is the expectation of what? How do you represent that formulaically?
expectation of the conditional expectation of Y given X; E(Y)=E[E(Y|X)]
What are the two main ways to determine whether or not X and Y are independent?
if P(x,y)=P(x)P(y) for every cell, then X and Y are independent; if P(y|x)=P(y) for all possible values of X and Y, then X and Y are independent
X and Y are uncorrelated if what?
if covariance (and correlation) between two random variables is 0
When is a random variable a continuous random variable?
if it can take on any value in an interval
When is a random variable a discrete random variable?
if it can take on no more than a countable number of values
How do you know if X and Y are independent?
if knowing the value of one of the variables provides no information about the other
What does a positive correlation indicate?
if one random variable is high, then the other random variable has a higher probability of being high, and we say that the variables are positively dependent
What does a negative correlation indicate?
if one random variable is high, then the other random variable has a higher probability of being low, and we say that the variables are negatively dependent
If X and Y are a pair of jointly distributed random variables, what is marginal probability distribution?
in this context the probability distribution of the random variable X is obtained by summing the joint probabilities over all possible values
What does a correlation of 0 indicate?
indicates that there is no linear relationship between 2 random variables
What is the probability distribution function P(x) of a discrete random variable X?
it represents the probability that X takes the value x, as a function of x
What is a random phenomenon?
it's a process leading to two or more possible outcomes, without knowing exactly which outcome will occur
How do you find standard deviation?
it's the positive square root of the variance
The probability distribution function is frequently referred to simply as the what?
probability distribution
If X and Y are a pair of jointly distributed discrete random variables, what is the conditional probability distribution of the random variable Y?
the conditional probability distribution of the random variable Y, given that the random variable X takes the value x, expresses the probability that Y takes the value y, when the value x is fixed for X
What is variance?
the expectation of the squared deviations about the mean
What is covariance?
the expected value of (X-µx)(Y-µy) is called the covariance between X and Y
The expected value of a random variable is also called what?
the mean of the random variable
Correlation measures what?
the strength of a linear relationship between two random variables
If X and Y are pair of discrete random variables, what does joint probability distribution express?
their joint probability distribution expresses the probability that simultaneously X takes the specific value x, and Y take the value y, as a function of x and y
If covariance and correlation between 2 variables does not equal 0, then what do we know about X and Y?
they are not independent
If X and Y are a pair of random variables with means µx and µy and variances σ²x and σ²y and X and Y are independent, what is the variance of their sum?
var (X+Y)=σ²x + σ²y
If X and Y are a pair of random variables with means µx and µy and variances σ²x and σ²y and X and Y are independent, what is the variance of their difference?
var(X-Y)=σ²x + σ²y
The mean of Y is the weighted average of _____, weighted by the _____
weighted average of the conditional expectation of Y given X, weighted by the probability distribution of X
When is correlation 0?
when 2 random variables are independent
In P(y=1|x=2), what is given and where does that value go in the conditional probability distribution?
x=2 is given and it goes in the denominator
What's the equation for conditional mean?
µy|x=E[Y|X]=∑(y|x)*P(y|x)
What is a more shortened version of the equation for correlation?
ρ=cov(x,y)/σxσy
What is the equation for variance?
σ²=E[(X-µx)²]=∑(x-µx)²P(x)
If X and Y are a pair of random variables with means µx and µy and variances σ²x and σ²y and Cov(X,Y)≠0, then what is var(X+Y)?
σ²x+σ²y+2ρxyσxσy
If X and Y are a pair of random variables with means µx and µy and variances σ²x and σ²y and Cov(X,Y)≠0, then what is var(X-Y)?
σ²x+σ²y-2ρxyσxσy
What is the equation for conditional variance?
σ²y|x=∑((y-µy|x)²|x)*P(y|x)
What is the formula for covariance?
∑∑(x-µx)(y-µy)*P(x,y)
What is the equation for correlation?
∑∑(x-µx/σx)(y-µy/σy)*P(x,y)
If 2 random variables are statistically independent, what is the covariance between them?
0
The probabilities must satisfy what two requirements?
0≤P(x)≤1 for any value of x; the individual probabilities sum to 1: ∑P(x)=1
What are the two requirements of the marginal probability distribution?
0≤P(x,y)≤1 for any pair of values x and y; the sum of the joint probabilities P(x,y) over all possible pairs of values must be 1
If X and Y are a pair of random variables with means µx and µy and variances σ²x and σ²y, what is the expected value of their sum?
E[X+Y] = μx + μy
If X and Y are a pair of random variables with means µx and µy and variances σ²x and σ²y, what is the expected value of their differences?
E[X-Y] = μx - μy
The expected value of a discrete random variable X is defined as what?
E[X]=µx=∑x*P(x)
what is covariance?
a measure of the direction of a linear relationship between two random variables, however it does not measure the strength of a linear relationship between two random variables
What is a random variable?
a variable that takes on numerical values realized by the outcomes in the sample space generated by a random phenomenon or random experiment
How do you calculate mean when random variable Y is a+bx?
a+bµx
Suppose W=aX+bY where a and b are constants, then µw equals what?
aµx + bµy
If E(Y|X)=E(Y), then what do Cov(X,Y) and ρxy equal?
both equal 0
A strong linear relationship is defined as what kind of condition?
condition where the individual observations are close to a straight line
If covariance=0, what does this tell us about correlation?
correlation=0; variables are uncorrelated
If Cov(X,Y)=0, this does not necessarily imply what?
does not imply that E(Y|X)=E(Y)
When covariance equals 0, this does not imply what?
does not necessarily imply that X,Y are independent
