Question

let the discrete random variable, x, have the following probability mass function

f(x)= c|x-2|, x=-2,-1,0,1,2,3

a. find the constant such that fx is a valid pmf

b. find P(|X-1|>1)

c. find the expected value of X

d. find the expected value of X^2

Answer 1

X	P(X)
-2	4c
-1	3c
0	0
1	1c
2	0
3	c
total	9c

a)  
total sum 9c must be 1 to be a valid PDF  
so, 9c=1  
c=1/9  
  
b)
P(|X-1|>1)

|X-1|	P(X)
3	=4/9
2	=3/9
1	=0
0	=1/9
1	0
2	=1/9

so, P(|X-1|>1 = P(|X-1|=2)+P(|X-1|=3) = 3/9+4/9=7/9

c)

X	P(X)	X*P(X)
-2	4/9	-0.88889
-1	1/3	-0.33333
0	0	0
1	1/9	0.111111
2	0	0
3	1/9	0.333333

	P(X)	X*P(X)
total sum =	1	-0.77778

mean = E[X] = Σx*P(X) =			-0.77778

d)

X²	P(X)	X²*P(X)
4	4/9	1.777778
1	1/3	0.333333
0	0	0
1	1/9	0.1111
4	0	0
9	1/9	1

	P(X)	X²*P(X)
total sum =	1	3.2222

E[X²] = Σx²*P(X) =            3.2222