Question

# Two balls are chosen randomly from an urn containing 6 red and 4 black balls, without...

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)

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

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