wannabe92 said:Hi, I've no idea where to go with the question below:
Joint moment generating function of X and Y - MXY(s,t) = 1/(1-2s-3t+6st)
for s<1/2, t<1/3.
Find P(min(X,Y) > 0.95) and P(max(X,Y) > 0.8)
Ray Vickson said:Can you represent [itex] M_{XY}(s,t)[/itex] in the form [itex] M_1(s) \cdot M_2(t)[/itex]?
RGV
wannabe92 said:Is it 1/(1-2s) . 1/(1-3t)?
A Joint Moment Generating Function (JMGF) is a mathematical function that characterizes the probability distribution of a joint random variable. It is defined as the expected value of the exponential of the joint random variable multiplied by a vector of constants. The JMGF can be used to find the moments of the joint random variable, which provide information about its central tendency and spread.
The Joint Moment Generating Function is important because it allows us to find the moments of a joint random variable, which provide information about its probability distribution. This can be useful in many statistical applications, such as hypothesis testing, parameter estimation, and confidence interval construction.
The Joint Moment Generating Function is calculated by taking the expected value of the exponential of the joint random variable multiplied by a vector of constants. This can be done using integration for continuous distributions or summation for discrete distributions. It is important to note that the JMGF may not exist for all joint random variables.
The Joint Moment Generating Function has several properties that make it a useful tool in probability and statistics. These include the fact that the JMGF uniquely determines the probability distribution of a joint random variable, it is a powerful tool for finding moments and generating moments of moments, and it is often easier to work with mathematically compared to the joint probability distribution function.
The Joint Moment Generating Function is closely related to other probability functions, such as the joint probability distribution function, the characteristic function, and the moment generating function. In fact, the JMGF can be seen as a generalization of the moment generating function to multiple random variables. Additionally, the JMGF can be used to find the characteristic function, which is useful for finding the probability density function and other properties of a joint random variable.