Scaling API-first – The story of a global engineering organization
Random variables
1. Random Variables
VOCABULARY
RANDOM VARIABLE
PROBABILITY DISTRIBUTION
EXPECTED VALUE
LAW OF LARGE NUMBERS
BINOMIAL DISTRIBUTION
BINOMIAL RANDOM VARIABLE
BINOMIAL COEFFICIENT
GEOMETRIC RANDOM VARIABLE
GEOMETRIC DISTRIBUTION
SIMULATION
2. Key Points
A random variable is a numerical measure(face up number of a
die) of the outcomes of a random phenomenon(rolling a die)
If X is a random variable and a and b are fixed numbers, then
μₐ₊ᵦₓ= a+βµₓ and Ợ²ₐ₊ᵦₓ=b²Ợ²x
If X and Y are random variables, then μₓ₊ᵧ= μₓ + μᵧ
If X and Y are independent random variables, then Ợ² ₓ₊ᵧ=
Ợ²ₓ + Ợ²ᵧ and Ợ² ₓ₋ᵧ= Ợ²ₓ + Ợ²ᵧ
As the number of trials in a binomial distribution gets
larger, the binomial distribution gets closer to a normal
distribution
5. Random Variable
A ______ ______ is a numerical measure of the
outcomes of a random phenomenon
The driving force behind many decisions in
science, business, and every day life is the
question, “What are the chances?”
Picking a student at random is a random
phenomenon.
The students grades, height, etc are random
variables that describe properties of the student.
8. Random Variable
The random variables can be categorical as well( top album, movies
watched, favorite artists, etc)
9. Random Variable- Probability distribution
A _______ ________ is a listing or graphing of
the probabilities associated with a random variable
10. Random Variable- Probability(or population)
distribution
The probability distribution can be used to answer
questions about the variable x( which in this case is the
number of tails obtained when a fair coin is tossed three
times)
Example: What is probability that there is at least one tails
in three tosses of the coin? This question is written as
P(X≥1)
P(X≥1)= P(X=1) + P(X=2)+ P(X=3)= 1/8 +3/8+3/8= 7/8
11. Random variable- discrete and continuous
_______ random variables takes a countable
number of values(# of votes a certain candidate
receives)
_______ random variables can take all the possible
values in a given range(the weight of animals in a
certain regions)
14. Random variable- expected value
The mean of the probability distribution is referred
to as the ______ ______, and is represented by
μₓ.
which just means that the mean(or expected value)
of a random variable is a weighted average
15. Random Variable- Expected Value
For this probability distribution, the
expected value is
= 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8)= 12/8=
1.5
16. Law of Large Numbers
The _______ of _______ _______states that the
actual mean of many trials approaches the true mean
of the distribution as the number of trials increases
18. Binomial Distribution
________ ________ models situations with the
following conditions:
1. Each observation falls into one of just two categories(
success or failure)
2. The number of observations is the fixed number n
3. The n observations are all independent
4. The probability of success, p, is the same for each
observation
19. Binomial Distribution
For data produced with the binomial model, the
binomial random variable is the number of
successes, X.
The probability distribution of X is a binomial
distribution
When finding binomial probabilities, remember that
you are finding the probability of obtaining k successes
in n trials
25. Geometric Distribution
Each observation falls into one of two categories,
success or failure
The variable of interest (usually X) is the number of
trials required to obtain the first success
The n observations are all independent
The probability of success, p, is the same for each
observation
26. Geometric Distribution
Example: If one planned to roll a die until they got a 5, the random
variable X= the number of trials until the first 5 occurs.
Find the probability that it would take 8 rolls given that all the
conditions of the geometric model are met
27. Geometric Distribution
Expected Value of Geometric Distributions
If X is a geometric random variable with probability of success P
on each trial, then the mean or _______ _______ of the
random variable is μ= 1/p.