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
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’
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