Final answer:
The constant c for the probability density function f(x) = cx on the interval [0,1] is found to be 2, ensuring the total area under the curve equals 1, which represents the total probability of all outcomes.
Step-by-step explanation:
The value of the constant c for the probability density function f(x) = cx, where x is in the interval [0,1], can be found by ensuring that the total area under the curve of f(x) over the interval [0,1] equals 1. This is because the area under a probability density function over its entire range must equal 1, as this represents the total probability of all outcomes.
To find the value of c, we integrate the function f(x) = cx over the interval from 0 to 1:
∫01 cx dx = 1
The integral of cx from 0 to 1 is (c/2)x2 evaluated from 0 to 1, which is c/2. Setting this equal to 1 gives us c/2 = 1. Solving for c gives us c = 2.
Therefore, the value of c that makes the function a probability density function is 2.