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Probabilities are used to determine the chances of an event.
(a) The expected value
This is the mean of the distribution, and it is calculated using:
[tex]\mathbf{E(x) = \sum x \cdot P(x)}[/tex]
So, we have:
[tex]\mathbf{E(x) = 25 \times 0.24 + 50 \times 0.38 + 75 \times 0.30 + 100 \times 0.08 }[/tex]
Using a calculator, we have:
[tex]\mathbf{E(x) = 55.5}[/tex]
Hence, the expected value is: 55.5
(b) The standard deviation
This is calculated using:
[tex]\mathbf{\sigma = \sqrt{E(x^2) - (E(x))^2}}[/tex]
Where:
[tex]\mathbf{E(x^2) = \sum x^2 \cdot P(x)}[/tex]
So, we have:
[tex]\mathbf{E(x^2) = 25^2 \times 0.24 + 50^2 \times 0.38 + 75^2 \times 0.30 + 100^2 \times 0.08 }[/tex]
[tex]\mathbf{E(x^2) = 3587.5 }[/tex]
So, we have:
[tex]\mathbf{\sigma = \sqrt{E(x^2) - (E(x))^2}}[/tex]
[tex]\mathbf{\sigma = \sqrt{3587.5 - 55.5^2}}[/tex]
[tex]\mathbf{\sigma = \sqrt{3587.5 - 3080.25}}[/tex]
[tex]\mathbf{\sigma = \sqrt{507.25}}[/tex]
[tex]\mathbf{\sigma = 22.52}[/tex]
Hence, the standard deviation is 22.52
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