# Chapter 9: Random variables

STAT 1010 - Fall 2022

# Learning outcomes

By the end of this lesson you should:

• Know how to find the mean given a probability mass or distribution function

• Understand how to use multiplication and addition rules to find expected values and variance of new random variables

# Revision - 1

Find the mean, and variance for these examples:

1. $20, 24, 25, 36, 25, 22, 23$

2. Rolling a fair six-sided die

# Revision - 2

A stock prices ($X$) at these frequencies:

$x$ $n$
1.23 3
1.29 5
1.37 4
1.84 1
1.18 6
1.22 2
1.25 4
Total 25

## Using proportions

$x$ $n$ $P(X) = x$
1.23 3
1.29 5
1.37 4
1.84 1
1.18 6
1.22 2
1.25 4
Total 25

## Mean or expected value

If we know $P(X) = x$, then we can use this to find the mean or expected value of a random variable. The pdf includes information about both the total and the number of occurrences of $x$, it does the computation for us.

$\mu = E(X)$

$= x_1p(x_1) + x_2p(x_2) + … + x_np(x_n)$

# Probability density function

There are different ways to define a pdf:

• table

• plot

# plot - pdf

## Input data
values <- c(1.23, 1.29, 1.37, 1.84, 1.18, 1.22, 1.25)
counts <- c(3, 5, 4, 1, 6, 2, 4)

d <-
as_tibble(data.frame(values, counts)) %>% # make tibble
mutate(prop = counts/sum(counts), # find proportion
part_mean = values*prop,  # multiply values & prop
mean = sum(part_mean)) # find the sum

d %>%
ggplot(aes(x = values, y = prop)) + # for values and prop
geom_bar(stat = "identity") + # make bar chart of identity (prop)
geom_vline(aes(xintercept = mean, color = "red")) # and line of mean

# Price increase

All the stock values have increased by \$5:

What happens to the mean?

What happens to the standard deviation?

## Price increase

$x$ $x + 5$ $P(X) = x+5$
1.23 6.23 0.12
1.29 6.29 0.20
1.37 6.37 0.16
1.84 6.84 0.04
1.18 6.18 0.24
1.22 6.22 0.08
1.25 6.25 0.16

## Price increase

$E(X \pm c) = E(X) \pm c$

$SD(X \pm c) = SD(X)$

$Var(X \pm c) = Var(X)$

# Stock splits

Without decreasing the cost, the stock values are now 3 times what they were before:

What happens to the mean?

What happens to the standard deviation?

## Stock splits

$x$ $3x$ $P(X) = x$
1.23 3.68 0.12
1.29 3.87 0.20
1.37 4.11 0.16
1.84 5.52 0.04
1.18 3.54 0.24
1.22 3.66 0.08
1.25 3.75 0.16

# Multiply by a constant

$E(cX) = cE(X)$

$SD(cX) = |c|SD(X)$

$Var(cX) = c^2Var(X)$

$E(cX \pm a)=cE(X) \pm a$

$Var(cX \pm a)=c^2Var(X)$

# Parameter vs estimate

population sample
name parameter estimate
mean $\mu$ $\bar{x}$
variance $\sigma^2$ $s^2$

standard

deviation

$\sigma$ $s$
size $N$ $n$