Question

Let \(f\left(x_{1}, x_{2}\right)=4 x_{1} x_{2}, 0

Short answer

The probabilities can be obtained by integrating the given function within the specific limits. To calculate them, perform double integration, keeping in mind that the limits for both integrals may change with each term.

Step-by-step solution

Finding P(0

To obtain this, we perform double integration on the given function within the given limits. The joint pdf $f(x_1,x_2)=4x_1{x}_2$ is used over the region $0<X_1<\frac{1}{2},\frac{1}{4}<X_2<1$. Using $x_2$ limits from 1/4 to 1 and $x_1$ limits from 0 to 1/2, we integrate the function. This gives us the desired probability.

Finding P(X_{1} = X_{2})

This involves evaluating the volume under the surface $f(x_1,x_2)=4x_1{x}_2$ and above the line segment $0<x_1=x_2<1$. For this, integrate the given function over the line $x_1=x_2$ from 0 to 1. This will give us the desired probability.

Finding P(X_{1} < X_{2})

To find this probability, we evaluate the joint pdf over the region where $x_1<x_2$. This will be the lower triangular region of the square [0, 1]x[0, 1]. Here, we carry out the double integration where $x_1$ varies from 0 to $x_2$ and $x_2$ varies from 0 to 1.

Finding P(X_{1} ≤ X_{2})

The probability $P(X_1\le X_2)$ can be found similarly to the previous step, except that the region of integration now includes the line $x_1=x_2$ i.e. the whole lower half of the [0,1]x[0,1] plane including the boundary. So, we need to do double integration where $x_1$ ranges from 0 to $x_2$ and $x_2$ ranges from 0 to 1.

Key concepts

Double integration

Double integration is a method used to calculate volumes under surfaces, which is particularly useful in finding probabilities when dealing with continuous random variables over two dimensions. Imagine you have a bumpy surface represented by a function, and you want to determine the volume enclosed by this surface over a defined area on the plane.In the exercise, the joint probability density function (pdf) is given as $f(x_1,x_2)=4x_1{x}_2$. To find the probability over a specific region such as \( 0

Identify the limits for $x_1$ and $x_2$.

Perform the integration first with respect to one variable, say $x_1$, over its limits while keeping $x_2$ constant.

Then, integrate the resulting expression with respect to $x_2$.

The order of integration can sometimes change depending on the region and function. For instance, you may integrate with respect to $x_2$ first, then $x_1$, depending on which simplifies the process. This double integration approach helps in computing the probabilities over a defined region under the surface.

Joint probability

Joint probability is the probability of two events happening at the same time. When dealing with continuous random variables, we model these events using a joint pdf, which in this case is represented by $f(x_1,x_2)=4x_1{x}_2$.In this context, imagine you have two variables, $X_1$ and $X_2$, which can take on any values between 0 and 1. The joint pdf describes how likely any combination of $X_1$ and $X_2$ is, as a function of $x_1$ and $x_2$.To calculate the probability of events such as $P(X_1=X_2)$, we need to think about where on the plane this equality holds. Essentially, this occurs along the diagonal line from (0,0) to (1,1). Using the joint pdf, we can find the volume under the surface along this line:

• Integrate the joint pdf along the line where $x_1=x_2$.
• This provides the probability of the two variables being equal over the given range.

Joint probability is a powerful concept because it allows us to understand the relationship between two random variables simultaneously. It is key in scenarios where the variables are dependent on each other.

Geometric probability

Geometric probability deals with the likelihood of an event happening within a given geometric region. It is particularly useful when dealing with continuous probability distributions where we can visualize outcomes in terms of areas and volumes.In the exercise, we are interested in probabilities such as \( P(X_{1}

For \( P(X_{1}

For $P(X_1\le X_2)$, it encompasses the entire lower half of the square, including the boundary line $x_1=x_2$.

To find these probabilities, we use the joint pdf to perform double integration over the regions defined by these conditions. This method allows for a geometric interpretation, translating areas under curves into probabilities for continuous distributions. Understanding geometric probability helps to bridge the concept of random variable relationships with visual and spatial reasoning.