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Lewis7879
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Does anyone know the proof of joint moment generating functions for bivariate normal distributions?
M_x,y (s,t)= E(e^(xs+yt))
M_x,y (s,t)= E(e^(xs+yt))
I'm actually suppose to do it likemathman said:Expand the exponential in a power series. E(each term) is that moment.
great ideamathman said:Expand the exponential in a power series. E(each term) is that moment.
No. [itex]e^{(xs+yt)}[/itex] must be under the integral sign.Lewis7879 said:I'm actually suppose to do it like
great idea
it is possible to do it this way ?
its a very long step.
M(s,t)= e(xs+yt) ∫∫ fXY (x,y) dxdy
= e(xs+yt) ∫∫ (1/(2π√(1-ρ2)σxσy)) exp [ [-1/2(1-ρ2)] [(x-μx/σx)2 + (y-μy/σy)2 - 2ρ(x-μx/σx)(y-μy/σy)}
I'm actually suppose to do it likemathman said:Expand the exponential in a power series. E(each term) is that moment.
yes I know that but will I get the solution of mgf of bivariate normal distribution ? I'm stuck current trying to derive it.mathman said:No. [itex]e^{(xs+yt)}[/itex] must be under the integral sign.
That will depend on what [itex]f_{XY}(x,y)[/itex] is. If it is bivariate normal, then you will get its moment generating function.Lewis7879 said:I'm actually suppose to do it like
M(s,t)=
yes I know that but will I get the solution of mgf of bivariate normal distribution ? I'm stuck current trying to derive it.
The bivariate normal distribution is a probability distribution that describes the joint distribution of two random variables. It is often used in statistical analysis to model the relationship between two continuous variables.
The moment generating function of a bivariate normal distribution is a mathematical function that allows us to calculate the moments of the distribution, such as the mean and variance. It is defined as the expected value of e^(tx), where t is a constant and x is a random variable.
The mgf of a bivariate normal distribution can be derived by using the definition of the moment generating function and the probability density function of the bivariate normal distribution. This involves integrating the probability density function over all possible values of the two random variables.
The moment generating function is important because it allows us to calculate the moments of the bivariate normal distribution, which are useful in statistical analysis. It also helps us to calculate other important characteristics of the distribution, such as skewness and kurtosis.
Yes, the mgf can be used to find probabilities for a bivariate normal distribution. By taking the inverse of the mgf, we can obtain the characteristic function, which can then be used to calculate probabilities for different values of the random variables.