Day10–Expected Value

Today’s Agenda

• Random Variables
• Expected Value
• Uniform Distribution

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Random Variable

• A random variable is a numerical outcome to a random process.

• eg. finding the sum of two dice, number of red crads drawn when drawing 5 cards, sampling random from a population produces random variables

• Most important example for us: measuring a quantitative summary (or characteristic) of a sample. The randomness comes from the possiblility of having selected a different sample.

• eg. A student has an assigned text book for a course. How much a random student in the course spends on the text book is a random variable, $X$.

• random variables are usually denoted with capitals $X$,$Y$,$Z$

• The book store assumes that students will either: not buy a book, buy the book used, or buy it brand new. The possible outcomes are usually denoted with lower case letters.

• Revenue for the book store is $x_1 = 0$, $x_2 = 137$, $x_3 = 170$ for the three possbilities and the book store believes they each occur with probability $.20$, $.55$, and $.25$ respectively. In probability notation:

$$P(X=0)=.20,P(X=137)=.55,P(X=170)=.25$$

• this is called a discrete probability distribution because there are only finitely many options.

books_data <- data.frame(prices = c("0","137","170"), probs = c(.2,.55,.25))
ggplot(books_data,aes(x=prices,y=probs)) + 
  geom_bar(stat="identity") +
  ylim(0,1) +
  labs(main="Proportion of students buying books at different prices")+
  ylab("Proportion")+ 
  xlab("Cost")

Another discrete probability distribution is that of rolling a die:

die <-data.frame(roll=c(1,2,3,4,5,6),probs=c(1/6,1/6,1/6,1/6,1/6,1/6))

ggplot(die,aes(x=roll,y=probs))+geom_bar(stat="identity") +
  ylim(0,1) + 
  scale_x_discrete(name ="Roll", 
                    limits=c("1","2","3","4","5","6"))

• When all probabilities are equally likely the distribution is called the uniform distribution.
• This particular uniform distribution is a discrete probability distribution.

Probability Distributions

• a probability distribution has two components:
  • all the probabilities must sum to $1$.
  • none of the probabilities can be negative.
• The expected value of a random variable with a uniform probability distribution is the mean of the possibilities

Probability Distribution

Add notes here

Expected Value

The expected value of a random variable is:

$$E(X) = x_1 \times P(X=x_1) + \dots + x_k \times P(X=x_k) = \sum_{i=1}^k x_i P(X = x_i)$$

and the standard deviation of the expected value of the random variable is

$$\sqrt{\sum (x_i-E(X))^2 P(X = x_i)}$$

If all the probabilities are the same, then $$E(X) = x_1 \times \frac{1}{k} + \dots + x_k \times \frac{1}{k} = \frac{\sum_{i=1}^k x_i}{k} = \bar{x}$$

this is just the mean.

eg. The expected price of a text book is:

0*(.2)+ 137 * (.55) + 170 *(.25)

## [1] 117.85

# the expected value of a students cost at the book store is $117.85

books_data_2 <- data.frame(prices = c(0,137,170), probs = c(.2,.55,.25))

expected_cost <- sum(books_data_2$prices * books_data_2$probs)

• The expected value is a just a generalization of the mean.
• The usual calculation for mean assumes every option is equally likely, the expected value does not.

A small example:

# 2, 3, 3, 4, 4, 4
# the old way
(2+3+3+4+4+4)/6

## [1] 3.333333

# The new way
(2*1/6) + (3*2/6)+(4*3/6) 

## [1] 3.333333

The standard deviation is:

sqrt(sum((books_data_2$prices - expected_cost)^2*books_data_2$probs))

## [1] 60.49238

# $60.49 is the standard deviation of the expected cost.

Continuous Distributions

• When data is numeric continuous, we can often make the bins of a histogram extremely small and still have a very readable graph

eg.

• the biggest difference between continuous and discrete probability distributions is that the y-axis represents a different thing
• for discrete, its the probability
• for continuous, its the density, and to find the probability you have to find the area.