Question

Before planting a crop for the next year, a producer does a risk assessment. According to her assessment, she concludes that there are three possible net outcomes: a 5,000 gain, a 6,500 ga 7,000 loss with probabilities 0.75, 0.15 and 0.1 respectively. The median profit is ________

Answer 1

Solution :

To obtain the median from a discreet probability distribution of X follow the given steps :

1) Make a cumulative probability distribution from the given probability distribution.

2) Find the largest value of X (say x_{​​​​​​1}) such that P(X ≤ x_{​​​​​​1}) ≤ 0.5.

3) Find the smallest value of X (say x_{​​​​​​2}) such that P(X ≤ x_{​​​​​​2}) ≥ 0.5.

4) Median = (x_{​​​​​​1} + x_{​​​​​​2})/2

Now the Cumulative probability distribution of the given probability distribution would be as follows :

x	P(X = x)	P(X ≤ x)
5000	0.75	0.75
6500	0.15	0.90
-7000	0.1	1.00

The largest value of X (say x₁) such that, P(X ≤ x_{​​​​​​1}) ≤ 0.5 is x_{​​​​​​1} = 5000.

The smallest value of X (say x_{​​​​​​2}) such that, P(X ≤ x_{​​​​​2}) ≥ 0.5 is x_{​​​​​​2} = 5000.

Hence,

Median = (5000 + 5000)/2

Median = 5000

The median profit is 5000.