Continuous Probability Distributions

By Nikki, Becky and Lauren

To get from F(x) to f(x) you must differentiate!

So split the fraction into twox + 4 . Then differentiate. 9 9

The equation for f(x) now is simply 1 There is no x here therefore it is9a rectangular distribution.f(x) = 1 (b – a)-4 0 5 x

19

f(x)

JAN 2007Rectangular Distribution

OR

This is simple in rectangular distribution as you can simply look at the graph and use the formula of (b x h)

Using the formulas on Page 11Mean = 1 (a +b) 2Variance = 1 (a – b) 12

2

Rectangular Distribution

• Use formulas from formula booklet.

The Expectation of a continuous random variableThe mean

score we would expect to obtain if samples were repeatedly taken from this distribution

Times by ‘t’

The cumulative distribution function

Medians, Quartiles and Percentiles

To plot the graph you must in put the values for x, in this case 0-4, in the equations given. For example when 0≤x≤1 you make x=1. ½(1) = ½, so you can plot this point onto the graph.

F(q₁) is equal to 0.25, as F(x)= ½ x for the lower quartile you have to times 0.25 by 0.5 to find the value of the upper quartile

Lower quartile = 0.25Median = 0.5

Upper quartile = 0.75

But the values 1.6 and 1.7 into this equation as this is where the upper quartile lies.

The upper quartile is equal to 0.75 so it does satisfy 1.6 q₃ 1.7.˂ ˂

Percentiles

• You many also be asked to find a percentile (e.g.10th)• The formula for this is similar to quartiles and

medians, the nth percentile F(pn)=n/100• If you were asked to find the 10th percentile you

would make F(p10)=10/100.

Formula PageProbability density Function: P(a<X<b)/P(a=X=b)=ʃ f(x) dx

Cumulative Distribution Function: F(x)=P(X=x)=ʃ f(x) dx

The Median, Quartiles and Percentiles:

(a) Median: F(m)=P(X=m)=ʃ f(x)dx=0.5(b) Lower quartile: =0.25(c) Upper quartile: =0.75

The Expectation of a Continuous Random Variable: E(X)=ʃ x f(x) dx

The Expectation of g(X): E[g(X)]= ʃ g(x) f(x) dx

The Variance of Continous Random Variable: Var(X)= E(X-E(X))² = E(X²)-[E(X)]²

The Standard Deviation of A Continuous Random Variable: VE(X²)-[E(X)]²

The Variance of a Simple Function of a Continuous Random variable: Var (a)=0 Var (aX)= a²Var(X) Var(aX+b)=a²Var(X)

x=b

x=a

x

d/dxF(x)=f(x)

m

all x

all x