Final answer:
The constant c for the probability density function f(x) = c(x+1) over the interval 0 to 5 is found by integrating the function and setting the total area equal to 1, which yields c ≈ 0.0571.
Step-by-step explanation:
Finding the Constant for a Probability Density Function
To find the value of c for the continuous random variable X with the density function f(x) = c(x+1) on the interval from 0 to 5, we will use the property that the total area under the curve of the probability density function (pdf) must equal 1. This is because the total probability of all outcomes for a continuous random variable is always 1. We perform the integration of the density function over the interval [0,5] to find c.
Step by step solution:
Set the integral equal to 1, because the total probability must equal 1.
Solve the equation for c to find the value that ensures the total area under the curve is 1.
Let's perform the integration:
Integration:
∫ f(x) dx = ∫ c(x+1) dx from 0 to 5 = [c(x^2/2 + x)] from 0 to 5
= c(5^2/2 + 5) - c(0^2/2 + 0)
= c(25/2 + 5)
= c(12.5 + 5)
= c(17.5)
Now we set the result equal to 1:
c(17.5) = 1
Solve for c:
c = 1/17.5
c = 1/17.5 = 0.0571 (rounded to four decimal places)
The value of c is approximately 0.0571.