Problem 3812: Find The Value Of C

Problem 3812let Ain Order To Find The Value Of C We Have To Int

Let, in order to find the value of c, we need to perform double integration over the specified domain using the appropriate limits of integration. The result of this integration should be set equal to 1 (since the total probability must be 1), and then we solve for c. The integration should be performed with respect to y first, as the limits are given for y in terms of x, and then with respect to x.

From the problem, the limits for y are from -x to x, and for x are from 0 to ∞. Performing the integral of the joint density function f(x, y) = c(x² - y²)e^(-x) over y from -x to x, we add the limits appropriately, leading to the calculation of c. Evaluating the integral, we find c = 1/8.

Next, to find the marginal density with respect to y, we integrate out x. Since the domain is given with y in terms of x such that y ≤ |x| and x ≥ 0, the limits for x are from |y| to ∞. The marginal density f_Y(y) is obtained by integrating the joint density over x within these limits.

Similarly, to find the marginal density of X, we integrate out y over the interval from -x to x, with the constant c determined earlier. This provides the marginal distribution of X.

Lastly, the conditional distribution of (X, Y) given Z = z involves computing the joint density conditioned on the event Z = z, which requires integrating the joint density over the appropriate domain and normalizing it.

Paper For Above instruction

The process of determining the value of the constant c in the joint probability density function involves an understanding of the properties of probability distributions, including integration over the domain to ensure the total probability equals one. As shown, integrating over the specified limits, first with respect to y and then with respect to x, allows us to calculate c. This approach hinges on correctly identifying the domain of the joint distribution and applying the limits accordingly.

Once c is obtained, the marginal densities are derived by integrating the joint density over the other variable across its respective domain. For example, integrating out x yields the marginal density of y, which captures the probability distribution of y irrespective of x. Conversely, integrating out y gives the marginal density of x.

The calculation of marginal densities is fundamental in probability theory because it simplifies complex joint distributions into simpler, univariate distributions. Properly executing these integrations involves understanding the structure of the domain, especially when the domain involves inequalities such as y ≤ |x|, which define the range of integration based on the other variable.

The conditional distribution of (X, Y) given Z=z further demonstrates applications of joint and marginal densities in defining conditional probabilities. This concept is central in Bayesian inference and statistical modeling when conditioning on observed data or parameters.

Overall, this problem illustrates core methods in probability theory for manipulating joint, marginal, and conditional distributions, emphasizing the importance of correct limits of integration based on the domain's geometric and algebraic constraints.

The applications of these techniques extend broadly, from statistical inference to reliability engineering, where understanding the distribution of variables conditioned on certain events aids in decision-making processes. Mastery of integration techniques and domain understanding is thus fundamental in advanced probability and statistics.

Conclusion

In conclusion, the calculation of the constant c, the derivation of marginal densities, and the understanding of conditional distributions form foundational components of probability theory. Accurate integration over the correct domains ensures the validity of the probability models, which are crucial in statistical analysis and modeling. Proper application of these principles enables statisticians and data scientists to analyze complex systems and make informed decisions based on probabilistic reasoning.

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