Section 4.1 - PDF
What are the two main characteristics of discrete PDFs (hint: what must be true about the probabilities)?
Each probability must be between 0 and 1, and the sum of the probabilities is one.
What is the Law of Large Numbers?
As the number of trials in an experiment increases, the difference between the theoretical probability and the relative frequency approaches 0.
What must the outcome be of a Binomial trial?
A success or failure.
What is a random variable?
a characteristic of interest in a population being studied, often denoted by a capital X
What is the formula for the mean of a discrete PDF?
Summation of x * P(x)
Suppose we want to measure the number of items a person buys at a particular grocery store. Assume all customers always buy between 0 and 40 items, inclusive. Can we use a discrete PDF to model this situation?
If P(X=0)=0.1, P(X=1)=0.05, P(X=2)=0.23, P(X=3)=0.22, P(X=4)=0, P(X=5)=0.4, what is E[X]?
E[X]=0.1*0+0.05*1+0.23*2+0.22*3+0*4+0.4*5=3.17
The outcome of a Binomial trial is independent. What does this mean?
The outcome of one trial does not predict the outcome of any other.
What is an expected value?
expected arithmetic average when an experiment is repeated many times
What is the formula for the expected value, E[X], of a discrete PDF?
Summation of x * P(x)
Suppose you measure your daily commute to HYSA to the nearest minute. If the probability that it is 0-10 minutes is 0, P(11)=0.01, P(12)=0.02, P(13)=0.04, and P(14)=0.11, what is P(X>14)?.
P(X>14)=0.82
Suppose you play a game that costs $3 to play. You spin a wheel where 20% of the time, the wheel lands on $0, 20% of the time, it lands on $2, and 60% of the time, it lands on $5. What is the long-term expected value of the game and will you, in the long run, expect to earn money from playing?
E[X]=3.4, you will expect to earn money! That's why this game will never exist in real life.
If 70% of people in a population like tacos, what is the probability that 5 or fewer people like tacos in a sample of 10?
P(X<=5)=0.1527
What is a Binomal probability distribution?
a discrete RV that arises from Bernoulli trials with notation X~Bin(n, p)
What is the formula for the standard deviation of X in a binomial distribution?
sqrt(npq)
If I am measuring the length of different pencils students are using in class to the nearest mm, is the P(X>160)=P(X>=160) assuming X is a RV representing the length of a student's pencil?
Not necessarily, since P(X=160) is not necessarily 0. Sometimes, P(X>=x)=P(X>x) for continuous PDFs, though.
Suppose you roll an unfair, weighted 6-sided dice such that the probability you land on a 1 is 0.48, a 6 is 0.48, and every other number is 0.01. What is E[X]?
If 9 U.S. state capitals (of 50 states) have populations of over 500,000, and we take a random sample of 7 capitals, what is the probability that exactly 2 from our sample have populations of over 500,000?
0.25225 (fun fact: Phoenix is the most populated state capital in the U.S.)
If P(X=0)=0.1, P(X=1)=0.05, P(X=2)=0.23, P(X=3)=0.22, P(X=4)=0, P(X=5)=0.4, what is P(2<=X<=4)?
P(2<=X<=4)=0.45
If P(X=0)=0.1, P(X=1)=0.05, P(X=2)=0.23, P(X=3)=0.22, P(X=4)=0, P(X=5)=0.4, what is the standard deviation of the variable?
Approximately 1.703
If 4,000 books in a library with 200,000 books have purple covers, and I choose 3 books at random, what is the probability that at least 1 book has a purple cover?
0.0588