Question

Two balls are chosen randomly from an urn containing 6 red and 4 black balls, without replacement. Suppose that we win $2 for each black ball selected and we lose $1 for each red ball selected. Let X denote the amount on money we won or lost.
(a) Find the probability mass function of X, i.e., find P(X = k) for all possible values of k.
(b) Compute E[X].
(c) Compute Var(X)

Answer 1

a) below is probability mass function of X :

P(X=-2)=P(both red balls)=(6/10)*(5/9)=1/3

P(X=1)=P(1red and 1 black ball)=2*(6/10)*(4/9)=8/15

P(X=4)=P(both black balls)=(4/10)*(3/9)=2/15

b)

x	P(x)	xP(x)	x2P(x)
-2	1/3	-0.667	1.333
1	8/15	0.533	0.533
4	2/15	0.533	2.133
	total	0.400	4.000
	E(x) =μ=	ΣxP(x) =	0.4000
	E(x2) =	Σx2P(x) =	4.0000
	Var(x)=σ2 =	E(x2)-(E(x))2=	3.840

E(X)=0.400

c)Var(X)= 3.840