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Let the random variable x represent the number of girls in a family with three children. Assume the probability of a child being a girl is 0.37. The table on the right describes the probability of having x number of girls. Determine whether the table describes a probability distribution. If it​ does, find the mean and standard deviation. Is it unusual for a family of three children to consist of three​ girls?

x p(x)
0 .250
1 .441
2 .259
3 .050

Respuesta :

LammettHash
All the probabilities add up to 1, so the table does indeed describe a distribution.

The mean is given by

[tex]\mathbb E(X)=\displaystyle\sum_x xp(x)=0\times0.250+1\times0.441+2\times0.259+3\times0.050=1.109[/tex]

The standard deviation is given by [tex]\sqrt{\mathbb E(X^2)-\mathbb E(X)^2}[/tex]. You have

[tex]\mathbb E(X^2)=\displaystyle\sum_xx^2p(x)=0^2\times0.250+1^2\times0.441+2^2\times0.259+3^2\times0.050=1.927[/tex]

so the standard deviation is approximately [tex]\sqrt{1.927-1.109^2}\approx0.835[/tex].

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