Question

A machine dispenses water into a glass. assuming that the amount of water dispensed follows a continuous uniform distribution from 10 ounces to 16 ounces, the average amount of water dispensed by the machine is

Answer 1

⇢
`{\boxed{\bold{Average \: amount = 13 \: ounces}}}`

Step-by-step-explanation:

Given question:-

⇢ A machine dispenses water into a glass. Assuming that the amount of water dispensed follows a continuous uniform distribution from 10 ounces to 16 ounces.

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Find:-

⇢ The average amount of water dispensed by the machine is

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Calculations:-

⇢ Let us assume 'a and b' as the first ounce and the second ounce.

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• 
  `\rm{a = 10 \: ounce}`
• 
  `\rm{b = 16 \: ounce}`

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⇢
`{\boxed{\bold{((a + b))/(2)}}}`

⇢
`\rm{((10 + 16))/(2)}`

⇢
`\rm{\cancel{(26)/(2)}}`

⇢
`{\boxed{\bold{13 \: ounces}}}`

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Therefore, 13 ounces is the average amount of water dispensed by the machine.

Answer 2

`f_X(x)=\begin{cases}\frac1{16-10}=\frac16&\text{for }10\le x\le16\\0&\text{otherwise}\end{cases}`


`\implies\mathbb E(X)=\displaystyle\int_(-\infty)^\infty xf_X(x)\,\mathrm dx=\int_(10)^(16)\frac x6\,\mathrm dx=13`