Download Worksheet #3

WORKSHEET #3
STA 291
Summer 2010
Discrete Random Variables
A random variable is a variable whose value is determined by a chance mechanism.
Note: The chance mechanism is formally described by a probability distribution of
the possible values of the random variable X, typically a table of probabilities.
A random variable is discrete provided that its possible values are countable.
𝑉𝑎𝑟(𝑋) = ∑(𝑥𝑖 − 𝜇)2 𝑃(𝑋 = 𝑥𝑖 )
𝐸 (𝑋) = ∑ 𝑥𝑖 𝑃(𝑋 = 𝑥𝑖 )
Example
A survey of Amazon.com shoppers reveals the following probability distribution of the number
of books purchased per hit.
X
P(X)
0
.48
1
.26
2
.14
3
.06
4
.03
5
.02
6
.01
Calculate the mean, variance, and standard deviation of the number or of books purchased per
hit.
P(X ≤ 2) =
P(1 < X < 6) =
Conditional Probability
P( A | B) 
P( A  B)
, provided P( B)  0
P( B)
Independence
𝑃(𝐴|𝐵) = 𝑃(𝐴)
OR
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴)𝑃(𝐵)
Additive Law of Probability
𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
Examples
In a high school, 53% of the students are female, 30% of the students are freshman, and 16% of
the students are freshman and female. What is the probability of randomly selecting a student
that is not a freshman?
Same scenario, what is the probability of randomly selecting a student that is either a freshman
or female?
If 35% of a group of people have blonde hair and 20% have both blonde hair and blue eyes then
what is the probability of someone having blue eyes if we already knew they had blond hair?
Binomial Distribution
The probability of observing k successes in n independent trials with the probability of success
being p is
𝑛
𝑃(𝑋 = 𝑘) = ( ) 𝑝𝑘 𝑞 𝑛−𝑘
𝑘
𝑛
𝑛!
( ) = 𝑘!(𝑛−𝑘)!
𝑘
The following formulas are for the mean and variance of a binomial distribution.
𝐸(𝑋) = 𝑛𝑝
𝑉𝑎𝑟(𝑋) = 𝑛𝑝𝑞
Examples
If event A has a probability of success of .71 and 7 experiments are run, find the following.
E(X) = _______________
Var(X) = _______________
P(X = 5) = _______________
P(X = 4) = _______________