Final answer:
The upper limit for the probability density function (pdf) of random variable Y is 4.
Step-by-step explanation:
The probability density function (pdf) is used to describe the probabilities for continuous random variables. The area under the density curve between two points corresponds to the probability that the variable falls between those two values. In this case, the pdf for random variable Y is given as fᵧ(y) = (3y²/64) when 0 ≤ y ≤ ?, which means that the pdf is valid for values of y between 0 and some upper limit. To find the upper limit, we need to set up and solve an equation.
Since the total area under the graph of fᵧ(y) is equal to 1, we can set up the following integral equation: ∫[0, ?] (3y²/64) dy = 1. To solve this equation, we need to evaluate the integral and solve for the upper limit ?.
∫[0, ?] (3y²/64) dy = [3/64]∫[0, ?] y² dy = [3/64] * [(y³/3)|[0, ?] = [(y³/64)|[0, ?] = (y³/64)|[0, ?],
where |[0, ?] denotes the evaluation of the expression at the upper limit of the integral. Now, we can set this equal to 1 and solve for ?:
(y³/64)|[0, ?] = 1 => ?³/64 = 1 => ?³ = 64 => ? = 4.
Therefore, the upper limit for the pdf of random variable Y is 4. This means that the pdf fᵧ(y) = (3y²/64) is valid for values of y between 0 and 4.