Question

The median value y of a continuous random variable is that value such that F(y)=0.5. Find the median value of the random variable in f(y)={ 1/8 y, 0≤y≤4

0, elsewhere φ 0.5 =

Answer 1

The median value y of a continuous random variable with the given probability density function f(y) is calculated by integrating f(y) to get the cumulative distribution function (CDF) and equating it to 0.5. Solving the resulting equation, we find that the median value of y is 2√2 or approximately 2.83.

Step-by-step explanation:

The median value y of a continuous random variable is that value such that F(y)=0.5. For the given probability density function f(y) = 1/8 y, with a range from 0 to 4, to find the median, we integrate f(y) from 0 to y and set the result equal to 0.5.

First, we find the cumulative distribution function (CDF), F(y), by integrating f(y):

1. Integrate f(y) = 1/8 y to get F(y) = (1/16)y².
2. Set F(y) equal to 0.5 and solve for y: (1/16)y² = 0.5.
3. Solve the quadratic equation to find y.

After solving, the median value y is found to be the positive square root of 8, which is 2√2 or approximately 2.83, within the domain of y.