Distribution summaries

Question 1

What is expectation

Answer 1

If X is a random variable with a range space R,

E [𝑋] = ∑ 𝑥𝑝(𝑥) where x is a set of the real numbers

The expectation is the probability-weighted sum over all possible outcomes of the random variable. Values with high probability contribute more to the sum than values with low probability, hence the informal interpretation of the expectation as a “typical value”.

Question 2

What is a moment

Answer 2

the R’th moment, of a random variable X is E[X^R]

• X can be discrete or continuous
• E[X^R] may not exist
• E[X^R] has some useful information to summarise our distribution of our random variable X

when r = 1 we get our expectation
when r = 2, the value is useful to get Variance and SD of X
When r = 3, the value is useful to get our skewness
when r = 4, the value is useful to get kurtosis

Question 3

What is a central moment

Answer 3

The R’th central moment of our random variable X, is E[(x- mu]^R]

where mu = E[X]

Question 4

What is the 1st central moment of a random variable X?

Answer 4

0

Question 5

What is the formula for variance

Answer 5

The second central moment, or

E[(x-mu)^2]

Question 6

What is the definition of variance

Answer 6

the spread of the distribution around the centre, which we have defined as the expectation

Question 7

What is the R’th standardised central moment?

Answer 7

E[((X-mu)/sigma)^R]

when R = 3, the skewness is defined

for symmetric distributions skew[x] = 0

Question 8

What is the skewness of a random variable X?

Answer 8

the 3rd central standardised moment of X

Question 9

In terms of probability distributions, what is a quantile

Answer 9

A quantile is a value in the range space of the random variable corresponding to a given exceedant probability

Question 10

let q be the set of whole numbers greater than one,The 𝑞-quantiles of a random variable 𝑋 is the set of quantiles of 𝑋 given by {𝑥1/𝑞,𝑥2/𝑞,…,𝑥(𝑞−1)/𝑞}.
The 𝑞-quantiles are the cut-points that divide the range space of 𝑋 into 𝑞 interva

Answer 10

let q be the set of whole numbers greater than one,The 𝑞-quantiles of a random variable 𝑋 is the set of quantiles of 𝑋 given by {𝑥1/𝑞,𝑥2/𝑞,…,𝑥(𝑞−1)/𝑞}.
The 𝑞-quantiles are the cut-points that divide the range space of 𝑋 into 𝑞 interva