Question

4. Consider the density function.

f(x)={k√x, 0 < x <1
={0, elsewhere.

a) Find k.
b) Find F(x) and use it to calculate P(0.3 < X < 0.6)

Answer 1

To find k, integrate the density function over its defined range and set it equal to 1. Then, integrate the density function to find the cumulative distribution function (CDF) and use it to calculate the desired probability.

Step-by-step explanation:

To find k, we need to calculate the area under the density function and set it equal to 1. Since the function is only defined from 0 to 1, the area under the curve is equal to the integral of k√x from 0 to 1. Integrating this function gives us [∫(0)^(1) k√x dx]. Integrate this expression and set it equal to 1:

1 = ∫(0)^(1) k√x dx

Solve this equation to find the value of k. Once you have the value of k, you can find the cumulative distribution function (CDF), F(x), by integrating the density function. Then, use the CDF to calculate P(0.3 < X < 0.6) by subtracting the cumulative probability at X = 0.3 from the cumulative probability at X = 0.6:

P(0.3 < X < 0.6) = F(0.6) - F(0.3)