Flashcards in L5 - Continuous Random Variables and Normal Distribution Deck (9)

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1

## What are Discrete Variables?

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- It takes specific, finite values within a range.

- P(X = a) is clearly identified and read on the vertical axis.

- Cumulative probabilities are calculated as sums.

2

## What are Continuous Variables?

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- It takes any value within a range.

- P(X = a) = 0.

- We identify the variable in

ranges and look at cumulative probabilities:

P(a < X < b)

P(X > a)

P(X < b)

3

## What is the Probability distribution for continuous variables called?

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- The probability distribution for a continuous variable X is called a probability density function (pdf), and it is denoted as p(X) or f(X).

- The pdf gives us the shape of the distribution, however remember that

for a continuous random variable the probability is NOT read on the vertical axis any longer.

- It is a cumulative probability and it is the shaded area under the pdf (shaped like a bell curve). This is calculated as the integral of the function p(X)

evaluated between a and b.(Or between - and b, or a and + ).

- The whole area under the pdf must integrate to 1

4

## What is Normal Distribution?

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- Perhaps the most famous, represents many physical phenomena.

- First proposed by Karl Frederick Gauss, as a model for the errors of

measurement that occur in calculating the paths of stellar bodies, hence

its alternative names, the Gaussian distribution and the ‘normal curve of

error’.

- Characterised by two parameters, its mean µ and variance σ^2

so

X~N(µ,σ^2)

- It is bell shaped and symmetric about µ: mean, mode and median are the

same. This shape is shown Mathematically:

p(X)= (1/sqrt(2πσ^2)) x exp[(-(X-µ)^2)/("σ^2)]

5

## Why can we use a table for Normal Distribution?

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-0 The normal distribution is invariant to scale transformation: if

X~N(µ, σ^2 ) and W = c+dX then W~N(c+dµ, d^2 σ^2).

- Thus if we set c = -µ/ σ and d = 1/ σ we can define this new variable:

- Z= X-µ/σ ~ N(0,1)

This is called a standard normal and its pdf p(Z) is defined as:

p(Z)= 1/sqrt(2π) x exp (-Z^2/2)

- Its probabilities have been tabulated by statisticians.

6

## What are some common formulae of Normal Distribution Statistical Table?

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- P(Z > z ) = 1-P(Z < z)

- P(a < Z < b) = P(Z < b) - P(Z < a)

-Due to the symmetry of the normal distribution

P(Z < -z) = P(Z > z) = 1- P(Z < z)

7

## How can you calculate probabilities for any Normally Distributed Variable?

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- Calculating probabilities for any normally distributed X can be easily done

by transforming X into Z.

We know that:

Z= X-µ/σ

then P(X < a) is the same as;

- P(Z < a-µ/σ)

We find the value:

z{a} = a-µ/σ

and then use the Z-probabilities to find P(Z < z{a})

8

## What is the critical Value?

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the value of Z that defines a certain probability i.e. the area in a certain tail

- in the critical value table the values of Z associated with different probabilities in the format P(Z > z) = α.

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