Math 488 Lecture 2, 3

These flashcards cover key concepts such as moments, skewness, and variations in statistics as discussed in Math 488 Lecture 2.

Kth Moment

A statistical measure of the shape of a distribution, specifically the Kth moment about the mean.

6th Central Moment

E[(x-x)⁶], a specific moment that provides insight into the distribution's shape.

Raw Moment

Moment about zero, which represents the average of the Kth power of the data.

Coefficient of Variation

A standardized measure of dispersion of a probability distribution, calculated as the ratio of the standard deviation to the mean.

Skewness

A measure of the asymmetry of the probability distribution of a real-valued random variable.

Positive Skewness

Occurs when the tail on the right side of the distribution is longer or fatter than the left side.

Negative Skewness

Occurs when the tail on the left side of the distribution is longer or fatter than the right side.

Symmetric Distribution

A distribution that is identical on both sides of its central point, with a skewness of 0.

Loss

The basis of a claim for damages under the terms of a policy.
-The monetary value of damage caused by an insurable event. Loss can

Skewness formula

μ³/σ³

Define Excess Loss RV

Let X be the Loss R.V for a given deductible with Pr(X>d) > 0. Yp = x-d given x > d and Yp>0. Yp is the amount of insurance payment.

d is what?

The amount of loss the insurance company is not paying (you pay)

Given a distribution of X, probability statements about Yp for a particular d can be made using rules of conditional probability. Fy(k) = what?

Fy(k) = Pr(Y < k) = Pr(x < d+k | x > d) which is the probability that the total loss is less than d + k given x > d

Using conditional probability formula Fy(k) = what?

Pr(d<= X <=d+k)/(Pr(X > d)) =
Pr(d<= X <=d+k)/(1-Fx(d))

E[Yp] = what? (Remember we can make conclusions on Yp based on conditional probability, remember to apply this principle to expectation)

E[x-d|x>d]

E[x-d|x>d] is what?

\int_{d}^{\infty}\!\left(x-d\right)f\left(x\right)\,dx / (1-Fx(d))

Kth moment formula

E[x^k] = M’k

Kth central moment formula

E[(x-μ)^k] = Mk

Write down discrete case for E[Yp]

Summation with xj > d

Survival function S(x)

S(x) = 1-Fx(x)

Represent E[Y^p] in terms of the survival function S(x) and S(y)

Define left censored and shifted loss random variable formula/piecewise definition

Let X be the loss R.V. For a given d with Pr(X<d), the left censored and shifted loss R.V is Y^L = X-d.
It’s 0 if X <= d
and X-d if X > d
-Accepts negative values and it is unconditional compared to excess loss R.V

Define left censored and shifted loss random variable in words

YL is the amount of payment when a loss is experienced even if a payment is not made. YL has a probability mass at 0 and a probability density above 0, it is a mixed distribution. The universe in this distribution is all losses

Formula related E[Y^L] to E[Y^p]

Fundamental identity for random variables: E[Z] = what integral

Fy(k) and f(y)k

Define Limited Loss and Right Censored R.V

Let X be a loss R.V for a given u (upper limit on x). We define Y = min[X, u] = X ^ u
Y= {x, x < u
u, x >=u

How is this R.V (Right Censored) complementary to the Excess Loss R.V

(x-d) + (x^d)=x
Pays you above d + pays you up to d = entire loss

In insurance, this complementary property means:

Buying one insurance policy with a limit of d and another with a deductible of d is equivalent to buying full coverage

With a limit of u (or d but write it with u), write the formula for E[Y]
Note that Y means Right censored and Yp is insurance payment

What is the alternative formula for expectation of right-censored E[Y]

E[Y] = Sx(x)dx

E[Y]+E[YL]

=E[X]

If 0<p<1, then the 100th percentile of the distribution of x is any value π*p such that

F(πp)= P(X<πp)<=p<=p(x<=πp)

and what is this formula for discrete distributions?

F(πp)= P(X<=πp)=p