Expected and Variance
Y = x + 3. If E[x] = 5, what is E[Y]?
8
Expected Value!
Mu/mean of binomial
trials(n) * probability(p)
What are the four qualifications X must have to be a binomial random variable?
1. two outcomes (usually success or failure)
2. fixed number of trials
3. trials are independent
4. fixed probability of success
Conditional Rule!
P(B|A) = P(A & B) / P(B)
Z = 3 x X. If E[X] = 5, what is E[Z]?
15
Variance!
The sum of (outcome - mean)^2 * (P[E])
Standard deviation of binomial
square root of np(1-p)
According to the National Organization for Rare Disorders (NORD), about 1 in 10 Americans suffer from a rare disease. If we let X = the number in a group of 10 randomly selected Americans with a rare disease, what kind of variable is X?
Binomial random variable
Addition Rule!
P(A or B) = P(A) + P(B) - P(A & B)
Consider Var[3*x]. If X=2.6, what is Var[Z]?
23.4
Standard Deviation!
Square root of Var
RStudio function that gives P(X=k) -> exactly k successes in n trials with probability of success = p
dbinom(k,n,p)
According to the National Organization for Rare Disorders (NORD), about 1 in 10 Americans suffer from a rare disease. If we let X = the number in a group of 10 randomly selected Americans with a rare disease, what is the value of n?
10
Multiplication Rule
P(A&B) = P(A) * P(B|A)
If the temperature in degrees Fahrenheit at a certain location is distributed with mean 68 degrees and a standard deviation of 4 degrees, what is the mean of the temperature in degrees Celsius at the same location? It is given that C = 5/9(F - 32).
E[C] = 20
If X has a high variance, what can we observe?
The values of X are a long way from the mean
RStudio function that gives P(X is less than or equal to k) -> gives the probability of k successes or fewer in n trials with probability of success = p
pbinom(k,n,p)
According to the National Organization for Rare Disorders (NORD), about 1 in 10 Americans suffer from a rare disease. If we let X = the number in a group of 10 randomly selected Americans with a rare disease, what is the value of p?
1/10 = 0.1
Bayes' Rule!
P(E|F) / sum of P(F)
For funnsies, what is the empirical formula?
frequency of E / number of trials
The mean describes the ___ of the distribution. The standard deviation describes the ___ of the distribution.
Center; spread
RStudio function that gives P(X > k)
1-pbinom(k,n,p)
According to the National Organization for Rare Disorders (NORD), about 1 in 10 Americans suffer from a rare disease. Simulate randomly choosing a group of 10 and determine the value of X, the number in the group with this rare disease?
rbinom(10,1,0.1) = 1 person out of 10
Complement Rule!
P(notA) = 1 - P(A)