Answer:
Given the coordinates of rectangle ABCD are;
A = (3, 5)
B = (5,5)
C = (5, 1) and
D = (3, 1)
The rule of rotation 90 degree counter-clockwise around the origin is given by;
Now, the coordinates of each vertex when rectangle ABCD is rotated 90 degree counter-clockwise around the origin are;
Answer:
a) c=3/4
b) for -1<x<1
F(x) = (3/4) ( x + (2-x^3)/3 )
c)
E(x)=0
Var(x) = 1/5
Step-by-step explanation:
Hi!
a)
In order to f(x) be a probability density function it must be normalized, which means that its integral must be equal to 1:
Since f(x) is zero for x>1 and x<-1
Therefore:
c=3/4
b)
The cumulative distribution fucntion F(x) can be obtained integrating f(x) from -∞ to x:
Since f(x) = 0 for x<-1
F(x) = 0 for x<-1
for -1<x<1:
for x>1
F(x)=1
c)
The mean E(x) can be found integrating xf(x)
We can easily infer that the mean must be zero, since f(x) is an even function and thus xf(x) is an odd function integrated in a simetric interval:
E(x) = 0
The variance of x, Var(x), can be evaluated integrating (x-E(x))^2f(x), since E(x)=0:
Since c=3/4
Var(x) = 1/5