Problem 13 - Applied Probability

Question

a. Sketch the region of definition and determine analytically the marginal density functions \(f_{X}\) and \(f_{Y}\). b. Use a discrete approximation to plot the marginal density \(f_{X}\) and the marginal distribution function \(F_{X}\) c. Calculate analytically the indicated probabilities. d. Determine by discrete approximation the indicated probabilities. \(f_{X Y}(t, u)=\frac{1}{8}(t+u)\) for \(0 \leq t \leq 2,0 \leq u \leq 2\) \(P(X>1 / 2, Y>1 / 2), \quad P(0 \leq X \leq 1, Y>1), P(Y \leq X)\)

Step-by-Step Solution

Answer

Marginal density functions are found by integrating the joint density over the other variable's range. Plots for marginal densities and distribution functions are done using discrete values. Exact probabilities are found by setting up the appropriate integrals, and approximate probabilities are found using discrete intervals.

Step 1 - Sketching the region of definition

The region of definition is the set of all points (t, u) where the joint density function is defined. For the given function, the region is a square with vertices at (0,0), (2,0), (2,2), and (0,2) since t and u are both constrained between 0 and 2.

Step 2 - Finding Marginal Density f_X(t)

The marginal density function f_X(t) is found by integrating the joint density function over the entire range of u:\[f_X(t) = \int_0^2 f_{XY}(t, u) \, du = \int_0^2 \frac{1}{8}(t+u) \, du\]. First, calculate the integral by treating t as a constant and integrating with respect to u.

Step 3 - Finding Marginal Density f_Y(u)

Similarly, to find f_Y(u), integrate the joint density function over the entire range of t:\[f_Y(u) = \int_0^2 f_{XY}(t, u) \, dt = \int_0^2 \frac{1}{8}(t+u) \, dt\]. Compute the integral by treating u as a constant and integrating with respect to t.

Step 4 - Plotting the Marginal Density f_X(t)

Use a discrete approximation to plot the marginal density f_X(t) by calculating the value of f_X(t) at several points within the region [0,2] and plotting these points on a graph.

Step 5 - Plotting the Marginal Distribution Function F_X(t)

The marginal distribution function F_X(t) is the cumulative distribution function and is found by integrating the marginal density function from the lower limit of t's range up to t: \[F_X(t) = \int_0^t f_{X}(s) \, ds\]. Calculate F_X(t) at discrete points within the region [0,2] and plot these points.

Step 6 - Analytical Calculation of Probabilities

Calculate each of the following probabilities analytically: \[P(X>1/2, Y>1/2), P(0 \leq X \leq 1, Y>1), P(Y \leq X)\] by setting up and evaluating the appropriate double integrals.

Step 7 - Discrete Approximation of Probabilities

For the discrete approximation, divide the range of X and Y into equal intervals and use the midpoint of each interval to approximate the probability of X and Y falling within a certain range. Sum the product of these probabilities by the area of each interval to find the approximate probabilities.

Key Concepts

Probability Density Function

When studying probability and statistics, one of the most fundamental concepts is the Probability Density Function (PDF). A PDF, represented as f(x), describes the likelihood of a continuous random variable to take on a specific value. Unlike a probability mass function for discrete variables, a PDF gives us probabilities for intervals rather than exact points. Since the exact probability of a continuous random variable taking on a single, precise value is zero, the PDF is used to determine the probability that a variable falls within a certain range.

Integration plays a crucial role in working with PDFs. For any continuous random variable X, the total area under the PDF curve over all possible values of X is equal to 1. This means that if you integrate the PDF over its entire range, you will get:
\[\int_{-\infty}^{\infty}f(x)dx=1\]

This property is fundamental as it assures that the PDF is properly normalized and represents a valid probability distribution. To find probabilities over a specific range, you integrate the PDF over that range.

Cumulative Distribution Function

Another key concept in understanding probability distributions is the Cumulative Distribution Function (CDF), F(x). The CDF of a random variable X gives the probability that X will take on a value less than or equal to x. Mathematically, it is defined by the integral of the probability density function up to x:
\[F(x) = \int_{-\infty}^{x}f(t)dt\]

The CDF is a non-decreasing function that approaches 0 as x approaches negative infinity and 1 as x approaches positive infinity. This makes it incredibly useful for determining probabilities for ranges of values. For instance, the probability that X is between a and b is found by:
\[P(a \leq X \leq b) = F(b) - F(a)\]
Furthermore, the slope of the CDF at a particular point is equal to the PDF at that point, illustrating the close relationship between these two functions.

Integration in Probability

The process of Integration in Probability underpins the calculation of probabilities for continuous random variables. It's through integration that one can find the area under the PDF curve, which corresponds to the probability of a random variable falling within a specific interval. For example, if we have a PDF f(x), and we want to calculate the probability that X falls between two values a and b, we would compute the integral:
\[P(a \leq X \leq b) = \int_{a}^{b}f(x)dx\]

This integration will yield the area under the curve from a to b, giving us the probability we seek. In applications, you may be required to evaluate more complex probabilities, such as those involving multiple variables. For instance, to compute joint probabilities for two random variables X and Y with joint density function f_{XY}(x, y), we would integrate over a region R in the xy-plane:
\[P((X,Y) \in R) = \int \int_R f_{XY}(x,y)dxdy\]

The principles of integration are thus essential to understanding and computing probabilities, applying not just to single variables, but also to multivariable cases, where double or even triple integrals may be utilized.