L5 - Continuous Random Variables and Normal Distribution

Question 1

What are Discrete Variables?

Answer 1

• 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.

Question 2

What are Continuous Variables?

Answer 2

• 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)

Question 3

What is the Probability distribution for continuous variables called?

Answer 3

• 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

Question 4

What is Normal Distribution?

Answer 4

• 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)]

Question 5

Why can we use a table for Normal Distribution?

Answer 5

-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.

Question 6

What are some common formulae of Normal Distribution Statistical Table?

Answer 6

• 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)

Question 7

How can you calculate probabilities for any Normally Distributed Variable?

Answer 7

- 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})

Question 8

What is the critical Value?

Answer 8

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.

Question 9

What is the process for calculating the value of X from a given probability?

Answer 9

• Using the Critical Value find the value of z for that given probability
• to then find the value of X use the formula Z= a-µ/σ where X=a –> a=z{a}σ+µ