Let Y1 denote the weight (in tons) of a bulk item stocked by...

Let $Y_{1}$ denote the weight (in tons) of a bulk item stocked by a supplier at the beginning of a week and suppose that $Y_{1}$ has a uniform distribution over the interval $0 \leq y_{1} \leq 1$. Let $Y_{2}$ denote the amount (by weight) of this item sold by the supplier during the week and suppose that $Y_{2}$ has a uniform distribution over the interval $0 \leq y_{2} \leq y_{1},$ where $y_{1}$ is a specific value of $Y_{1}$ a. Find the joint density function for $Y_{1}$ and $Y_{2}$ b. If the supplier stocks a half-ton of the item, what is the probability that she sells more than a quarter-ton? c. If it is known that the supplier sold a quarter-ton of the item, what is the probability that she had stocked more than a half-ton?

Let $Y_{1}$ denote the weight (in tons) of a bulk item stocked by a supplier at the beginning of a week
and suppose that $Y_{1}$ has a uniform distribution over the interval $0 \leq y_{1} \leq 1$. Let $Y_{2}$ denote the amount (by weight) of this item sold by the supplier during the week and suppose that $Y_{2}$ has a uniform distribution over the interval $0 \leq y_{2} \leq y_{1},$ where $y_{1}$ is a specific value of $Y_{1}$
a. Find the joint density function for $Y_{1}$ and $Y_{2}$
b. If the supplier stocks a half-ton of the item, what is the probability that she sells more than a quarter-ton?
c. If it is known that the supplier sold a quarter-ton of the item, what is the probability that she had stocked more than a half-ton?

Key Concepts

Integration Techniques in Probability

Integration techniques are fundamental in probability theory, especially when dealing with continuous random variables. They are used to compute probabilities by integrating a density function over a specified range or region. This technique becomes particularly important when calculating probabilities, marginal distributions, or expected values in problems where the density functions are defined over continuous intervals.

Marginal Distribution

The marginal distribution is the probability distribution of a subset of a collection of random variables, found by integrating the joint density function over the range of the other variables. This concept is useful for isolating the behavior of one variable from a joint distribution that involves several variables.

Uniform Distribution

A uniform distribution is one in which every outcome in a given interval is equally likely. For a continuous uniform distribution, the density is constant over the specified interval. This concept is applied when the random variables are stated to have a uniform distribution over their respective intervals, ensuring that the probability of any subinterval is proportional to its length.

Joint Density Function

A joint density function describes the probability distribution of two continuous random variables simultaneously. It is a function that, when integrated over a specific region, gives the probability that the random variables fall within that region. In problems involving dependency between random variables, as in this case, the joint density is often constructed by multiplying the marginal density of one variable by the conditional density of the other variable given the first.

Conditional Distribution

A conditional distribution represents the probability distribution of one random variable given that another random variable is known to have a specific value. This concept is crucial when a random variable's range or behavior depends on the value of another variable. It involves defining a conditional density function that reflects the dependency between the two variables.

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