Let X be the continous random variable with pdf given by:

Question

f(x)={0.09375(4-x²),-2<or=x<or=2

0, elsewhere

Calculate :

a.P(-1<x<1)

b.P(X<0.5 or X>0.5)

Answer 1

The area under the curve = 1 the sum total of all probabilities.

For a continuous random variable, summation is replaced by integration.

(a) P(-1<x<1)=0.09375[4x-x^3/3]{-1,1}=0.09375(4-1/3-(-4+1/3)=0.09375(8-2/3)=0.6875. Note that because of symmetry this is the same as 2*0.09375[4x-x^3/3]{0,1}.

(b) 1, because this covers the whole of x between -2 and 2. The area to the left of x=0.5 and the area to the right include all x. So x has to be in one or other of the areas.