Question

Suppose the cumulative distribution function of the random variable X is

F(x) = 0, x < -0.8
0.25x + 0.2, -0.8 < X < 3.2
1, 3.2 < x
Round your answers to 3 decimal places. Determine the following:
a) P(X < 1.8)

Answer 1

To calculate P(X < 1.8) given the CDF of the random variable X, we substitute 1.8 into the function F(x) = 0.25x + 0.2, yielding a probability of 0.650.

Step-by-step explanation:

The question asks us to calculate P(X < 1.8), given the cumulative distribution function (CDF) of the random variable X. The cumulative distribution function for X is defined in three parts:

F(x) = 0 for x < -0.8

F(x) = 0.25x + 0.2 for -0.8 < X < 3.2

F(x) = 1 for 3.2 < x

For continuous random variables, such as X in this problem, the probability of a single point P(X = c) is 0. Therefore, to find P(X < 1.8), we will use the function for the range where 1.8 falls, which is the second part of the function since -0.8 < 1.8 < 3.2.

Substitute x = 1.8 into the CDF equation: F(1.8) = 0.25(1.8) + 0.2

Calculate the value: F(1.8) = 0.45 + 0.2

Combine terms: F(1.8) = 0.65

Therefore, the probability P(X < 1.8) is 0.650 when rounded to three decimal places. This is the answer to the question asked, relating to the cumulative distribution function for the given random variable X.