Question

. Let x be a variable whose distribution is given by f(x) = c(1 − x), −1 < x < 1.(a) What is the value of c?(b) What proportion of the population has an x-value that is between 0 and 0.5?(c) Find the mean of x.(d) Find the standard deviation of x

Answer 1

a) for this to be valid": f(x)dx=1

f(x)dx =   c*(1-x) dx =c*(x-x²/2)|¹_-1 =c*2 =1

c=1/2=0.5

b)

proportion of the population has an x-value that is between 0 and 0.5 =P(0<X<0.5)= f(x)dx =   0.5*(1-x) dx =

0.5*(x-x²/2)|^0.5₀ =0.1875

c)

E(X)= xf(x)dx = 0.5*(x-x²) dx =0.5*(x²/2-x³/3)|¹_-1 = -1/3 =-0.3333

d)

E(X²)= x²f(x)dx = 0.5*(x²-x³) dx =0.5*(x³/3-x⁴/4)|¹_-1 =1/3 =0.333

hence Var(X)=E(X²)-(E(X))² =(1/3)-(-1/3)² =(1/3)-(1/9)=(2/9)

std deviation =sqrt(2/9)=√2/3 =0.471405