Question

The length (in centimeters) of a petal on a certain flower is a random variable with probability density function defined by f(x)=1​/sqrt8x​ for x in [ 1,25 ]. a. Find the expected petal length

Answer 1

The expected petal length is 1 centimeter.

In this case, the PDF is defined as:

`$f(x) = (1)/(2√(x))$` for
`$x$` in
`$[9, 16]$`

To find the expected petal length, we need to calculate the integral of
`$x \cdot f(x)$` over the interval
`$[9, 16]$`.

Let's set up the integral and solve it step by step:

`$\text{Expected petal length} = E(X) = \int_(9)^(16) x \cdot f(x) \, dx$`

`$E(X) = \int_(9)^(16) x \cdot (1)/(2√(x)) \, dx$`

To simplify the integral, we can rewrite the expression as:

`$E(X) = (1)/(2) \int_(9)^(16) x^{-(1)/(2)} \, dx$`

Next, we can use the power rule of integration to solve the integral:

`$E(X) = (1)/(2) \left[2x^{(1)/(2)}\right]_(9)^(16)$`

`$E(X) = \left[x^{(1)/(2)}\right]_(9)^(16)$`

Now, substitute the upper and lower limits into the expression:

`$E(X) = \left[√(16) - √(9)\right]$`

`$E(X) = \left[4 - 3\right]$`

`$E(X) = 1$`