91Ó°ÊÓ

The concept of the Cumulative Distribution Function (CDF) helps us understand the probability that a random variable takes a value less than or equal to a given number. This is quite handy for estimating results based on a distribution.
For a continuous random variable like in our problem, the CDF, denoted by \( F(y) \), is found by integrating the probability density function from the lower bound up to a value \( y \). This gives us a function that increases smoothly from 0 to 1 as \( y \) goes from the lower to the upper limit of its range.
In this exercise, the CDF \( F(y) \) was calculated by integrating the given density function \( f(y) = \frac{3}{2} y^2 + y \) from 0 to the value \( y \). This integration results in \( F(y) = \frac{1}{2}(y^3 + y^2) \).
This function tells us that for each value of \( y \) within the interval [0,1], we can determine the probability that the random variable is less than or equal to \( y \). For example, \( F(1) \) gives a result of 1, indicating that the probability is 100% for the variable to be any value equal to or less than 1.

The Normalization Condition is a fundamental property of probability density functions. It states that the total area under the curve of the probability density function must equal 1.
This is because the total probability of all possible outcomes must be 1, as something must happen in terms of the random variable.
In the current exercise, to meet the Normalization Condition, we calculated the integral of the function \( f(y) = cy^2 + y \) from 0 to 1 and set it to equal 1. Through integration, \( \int_{0}^{1} (c y^2 + y) dy = \frac{c}{3} + \frac{1}{2} \). This equation helped us find the constant \( c \), which was solved to be \( \frac{3}{2} \).
Ensuring this condition is satisfied is essential because it validates that our density function properly represents a probability model.

Conditional Probability involves computing the probability of an event given that another event has occurred. This is especially useful when dealing with overlapping probabilities or when specific conditions affect the likelihood of outcomes.
In our scenario, we were asked to find the probability that a student requires at least 30 minutes to finish the exam, given they have already taken at least 15 minutes. This is represented by \( P(Y \geq 0.5 \, | \, Y \geq 0.25) \).
This requires computing \( P(Y \geq 0.5) \, \) and \( P(Y \geq 0.25) \, \) using the CDF \( F(y) \), and then applying the formula for conditional probability:

• \( P(A \mid B) = \frac{P(A \cap B)}{P(B)} \).

We calculated \( P(Y \geq 0.5) = 1 - F(0.5) \) and similarly for \( P(Y \geq 0.25) \), and then divided the probabilities accordingly. This process helps us understand how existing conditions or constraints impact the outcome.

Calculating probability involves determining how often a certain event is expected to occur within the context of a probability model.
In continuous distributions, this can involve using both the probability density function and the cumulative distribution function. In our example, to find the probability that a student completes the exam in less than 30 minutes, we determined \( F(0.5) \).
By integrating up to 0.5, we find \( F(0.5) \) using the cumulative function: \[ F(0.5) = \frac{1}{2}(0.5^3 + 0.5^2) = 0.1875 \]. This result means there's an 18.75% chance a student finishes in under 30 minutes.
Probability calculations can vary in complexity, but generally involve these core tools to provide insights on expected outcomes, helping in planning and decision-making across different fields.