Chapter 4: Probability Distributions

Findley & Nguyen (2020), University of Illinois Urbana-Champaign Page 51

Chapter 4: Probability Distributions

Probability Distributions Probability Distribution: Possible values for a variable expressed alongside the probability of that value

taking place in a randomly selected observation. Discrete variables with a handful of definable categories might have their distribution represented in a

table. A continuous variable (or discrete with many categories) would need a density curve.

Example: Let X = the outcome of a fair, six-sided die. What is the probability distribution of X? X P(X) 1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6

Discrete Probability Distributions

The probability distribution of a discrete random variable can be represented visually or in a table. The example below shows the number of nights in the hospital that a random selection of 373 patients

needed after having a knee replacement.

From the bar plot above, we can approximate what percentage of time patients needed for

inpatient recovery. Let N be the number of nights in the hospital that a patient needs for inpatient recovery.

What is the P(N = 2)? What is the P(N > 3)?

°I ( x =D a- to

① ↳ prob . that the outcome of the die =/ is I .Note: Sum of probabilities of all possible outcomes= I

.

= I

0

00I(w =2) =

# patients stayed 2 nights * (w > 3) = #patientsstayed>3migh- # total# total patients

got 8 t 3

= }}÷ = O. 3619 =-- 0.2172

373

Chapter 4: Probability Distributions

Findley & Nguyen (2020), University of Illinois Urbana-Champaign Page 52

Practice: Let X = the number of pizza Alex orders per week. What is the probability Alex orders 4 pizzas this week?

What is the probability Alex orders more than 3 pizzas this week?

What is the probability Alex orders at most 2 pizzas this week?

What is the probability Alex orders at least 1 pizza this week?

Expected Value: the expected value of a trial is the average of all outcomes of the trials if we are to repeat the trial many times (think 100,000 or 1 million times!). Notation: expected value of X is E(X). Formula: 𝜇 𝐸 𝑋 𝑥1𝑃 𝑋 𝑥1 𝑥2𝑃 𝑋 𝑥2 ⋯ 𝑥𝑛𝑃 𝑋 𝑥𝑛

Practice: What is the expected number of pizza Alex orders per week?

Variance: a measurement of how spread out the outcomes of the trial are! (just like variance of data that we learned in Unit 1) Notation: variance of X is Var(X). Formula: 𝜎2 𝑉𝑎𝑟 𝑋 𝑥1 𝜇 2𝑃 𝑋 𝑥1 ⋯ 𝑥𝑛 𝜇 2𝑃 𝑋 𝑥𝑛

Standard Deviation: another measurement of the variability in the distribution. Notation: standard deviation of X is SD(X). Formula: 𝜎 𝑉𝑎𝑟 𝑋

Practice: What is the variance and the standard deviation of the number of pizza Alex orders per week?

X P(X) 0 0.05 1 0.1 2 0.2 3 0.35 4 5 0.1 o. ¥.

iii"

:c: ::::: :::::::= t

= O.2 t O

.I =⑧

I ( X f 2) = I C X -- O)tix) t I ( X -- 2) = 0.05 t O. I t O. 2= 0.350

I ( X Z l) = I - I ( X -- O) = I - O . 05=0.950