Flashcards in L3 - Random Variables - Discrete Deck (8)

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1

## What is a random variable?

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A random variable is the result of a chance situation, so its value depends on chance: we have a range of possible outcomes and a set of

probabilities associated to them. For example:

- The number of customers that will enter a shop in the next hour;

- The number of children in a family;

- The height of people;

- The number I get when throwing a die

The numerical outcome of a random variable can be either discrete or

continuous

2

## How is a Discrete Random Variable defined?

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- Discrete r.v.

- only finite values within a range, usually integers

(ex. the die). The sample space is defined over a finite number of

outcomes.

3

## How is a Continuous Random Variable defined?

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- Continuous r.v

- any value within a range, so the number of values is virtually infinite. The sample space has an infinite number of outcomes.

- Height or temperature which can be measured to any degree of accuracy;

but also the number of unemployed people in a country, or the expenditure

of a family in a year

. - Although finite and not divisible, the number of values the variable can take on is so large that for all practical purposes it can be

considered continuous.

4

## How is a Probability distribution defined?

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a list of all the possible outcomes of a discrete r.v. and their associated probabilities.

- using X for the outcomes and P(x) for their respective probabilities

- ΣP(x) = 1

5

## For Discrete random variable what is its probability when its represented on a graph?

### - the probability of an outcome is exactly identified and can be read on the vertical axis

6

## What is Cumulative Probability?

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- The cumulative probability is the probability that the variable lies within a certain range or is larger or smaller than a certain values.

- Cumulative probabilities are calculated as sums:

- F(x) = P(X ≥ x{a}) = Σ_(X ≥ x{a}) (P(x)) or 1- P( x < x{a})

- F(x) = P(X ≤ x{a}) = Σ_(X ≤ x{a}) (P(x))

- F(x) = P(X{a} ≤ X ≤ X{b}) = Σ_X{a}^X{b} (P(x))

7

## How do you find the Mean of a Random Variable?

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- The mean of a r.v. X is defined as the “expected value of X"

- E(X) = μ = ΣxP(x)

- In the sample you had x(bar) = Σx/n = Σx(1/n)

- concept is the same since the same P(x) = 1/n

- μ and x(bar) are two different things; μ is a parameter of the population,

so it refers to the “true” distribution of X. The sample mean is an

estimator, a statistic from a sample of the population (which is often all one has). Different samples will have different estimates.

- The larger the sample is the closer x(bar) will be to μ. This is because the sample mean is a consistent estimator of the population mean.

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