Question

On hot, sunny, summer days, Jane rents inner tubes by the river that runs through her town. Based on her past experience, she has assigned the following probability distribution to the number of tubes she will rent on a randomly selected day. x 25 50 75 100 Total P(x) .24 .38 .30 .08 1.00 (a) Calculate the expected value and standard deviation of this random variable X. (Round your answers to 2 decimal places.)

Answer 1

Probabilities are used to determine the chances of an event.

• The expected value is: 55.5
• The standard deviation is 22.52

(a) The expected value

This is the mean of the distribution, and it is calculated using:

$\mathbf{E(x) = \sum x \cdot P(x)}$

So, we have:

$\mathbf{E(x) = 25 \times 0.24 + 50 \times 0.38 + 75 \times 0.30 + 100 \times 0.08 }$

Using a calculator, we have:

$\mathbf{E(x) = 55.5}$

Hence, the expected value is: 55.5

(b) The standard deviation

This is calculated using:

$\mathbf{\sigma = \sqrt{E(x^2) - (E(x))^2}}$

Where:

$\mathbf{E(x^2) = \sum x^2 \cdot P(x)}$

So, we have:

$\mathbf{E(x^2) = 25^2 \times 0.24 + 50^2 \times 0.38 + 75^2 \times 0.30 + 100^2 \times 0.08 }$

$\mathbf{E(x^2) = 3587.5 }$

So, we have:

$\mathbf{\sigma = \sqrt{E(x^2) - (E(x))^2}}$

$\mathbf{\sigma = \sqrt{3587.5 - 55.5^2}}$

$\mathbf{\sigma = \sqrt{3587.5 - 3080.25}}$

$\mathbf{\sigma = \sqrt{507.25}}$

$\mathbf{\sigma = 22.52}$

Hence, the standard deviation is 22.52

Read more about expected values and standard deviations at:

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