# STAB22H3 Chapter Notes - Chapter 16: Random Variable, Squared Deviations From The Mean, Normal Distribution

CHAPTER 16 - RANDOM VARIABLES

WHERE ARE WE GOING?

- random variable - used to model probability of outcomes

- helps us discuss about, & predict random behaviour

MAIN TEXT

[1]

(ex)

- insurance company offers "death and disability" policy

- pays $10,000 when you die

- pays $5,000 when you are permanently disabled

- charges $50/yr for having this benefit

Q: is company likely to make profit selling this plan?

- to ans., need to know probability that clients will die/be disabled in any yr

- using thta actuarial info, company can calc. expected val. that it gains from this

policy

WHAT IS AN ACTUARY?

- ppl who estimate likelihood & costs of rare events in order for them to be insured

- req. financial, statistical and business skills

p423

EXPECTED VALUE: CENTRE

[1]

RANDOM VARIABLE - var. that has its numerical val. based on outcome of random

event

- denoted with X

(ex) amt the company pays out on an individual policy

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NOTATION ALERT

- X, Y, Z are most common letters used for random var's

- CAPITAL LETTERS

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- denote particular val has by using "x"

- ex. for insuracnce company, x = 10,000 (if you die this yr), $5000 (if disabled),

or $0 (if neither occurs)

DISCRETE RANDOM VARIABLE

CONTINUOUS RANDOM VARIABLE

- random var. s.t. we can list all of its

outcomes

(ex) x = 10,000 or 5000, or 0 are its

outcomes for insurance company ex.

- random var s.t. you cannot list all of its

outcomes (has too many)

PROBABILITY MODEL (for a random var.) = collection of all possible val's and

probabilities that they occur at

[2]

probability model for insurance company example

Policyholder

Outcome

Payout

(x)

Probability

P(X = x)

Death

10,000

1/1000

Disability

5,000

2/1000

Neither

0

997/1000

- suppose that company ensures 1000 ppl

- 1 policyholder dies

- 2 are disabled

- remaining 997 live the year unharmed

=> has to pay $10k to one client (die), and $5,000 to each of the 2 clients

=> 20k in total, or an average:

- $20,000 / 1000 policies = $20/policy

- expected value E(X)

- recall: company charges $50 for this policy

- then, in a year it loses $20/policy, it has made profit of $30 per policy

[3]

- cannot predict what WILL happen in any given yr, but can say what we EXPECT to

happen (E(X))

- req. probability model

- parameter of probability model is EXPECTED VALUE

- E(X) or μ

- which is mean