Random Variables - Distribution and Expectations

  • Thread starter topgun08
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Here is the homework question. I only have an issue with part c but have shown all my work up until then. Any help is appreciated!

Mr and Mrs Brown decide to continue having children until they either have their first boy or until they have
five children. Assume that each child is equally likely to be a boy or a girl, independent of all other children,
and that there are no multiple births. Let B and G denote the numbers of boys and girls respectively that the
Browns have.
(a) Write down the sample space together with the probability of each sample point.
Sample Space with probability = {B 1/2, GB 1/4, GGB 1/8, GGGB, 1/16, GGGGB 1/32, GGGGG 1/32}
(b) Write down the distributions of the random variables B and G.
Pr[B=0] = 1/32, Pr[B=1] = 31/32
Pr[G=0] = 1/2, Pr[G=1] = 1/4, Pr[G=2] = 1/8, Pr[G=3] = 1/16, Pr[G=4] = 1/32, Pr[G=5] = 1/32,

(c) Compute the expectations of B and G using a direct calculation
E(B) = 31/32

Is this below part correct. For some reason I don't think it could be correct.
E(G) = 31/32??????
 

Answers and Replies

  • #2
chiro
Science Advisor
4,790
132
Here is the homework question. I only have an issue with part c but have shown all my work up until then. Any help is appreciated!

Mr and Mrs Brown decide to continue having children until they either have their first boy or until they have
five children. Assume that each child is equally likely to be a boy or a girl, independent of all other children,
and that there are no multiple births. Let B and G denote the numbers of boys and girls respectively that the
Browns have.
(a) Write down the sample space together with the probability of each sample point.
Sample Space with probability = {B 1/2, GB 1/4, GGB 1/8, GGGB, 1/16, GGGGB 1/32, GGGGG 1/32}
(b) Write down the distributions of the random variables B and G.
Pr[B=0] = 1/32, Pr[B=1] = 31/32
Pr[G=0] = 1/2, Pr[G=1] = 1/4, Pr[G=2] = 1/8, Pr[G=3] = 1/16, Pr[G=4] = 1/32, Pr[G=5] = 1/32,

(c) Compute the expectations of B and G using a direct calculation
E(B) = 31/32

Is this below part correct. For some reason I don't think it could be correct.
E(G) = 31/32??????
Hey topgun08 and welcome to the forums.

Assuming your distribution for G is correct in part b, your answer is also correct. I used the calculation:

E[G] = 0x1/2 + 1x1/4 + 2x1/8 + 3x1/16 + 4x1/32 + 5x1/32 = 0.96875 = 31/32
 

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