The joint moment generating function of two random variables...

The joint moment generating function of two random variables $X$ and $Y$ is defined to be the function $M(s, t)$ of two real variables defined by $$ M(s, t)=\mathbb{E}\left(e^{s X+t Y}\right) $$ for all values of $s$ and $t$ for which this expectation exists. Show that the joint moment generating function of a pair of random variables having the standard bivariate normal distribution (6.73) is $$ M(s, t)=\exp \left[\frac{1}{2}\left(s^{2}+2 \rho s t+t^{2}\right)\right] $$ Deduce the joint moment generating function of a pair of random variables having the bivariate normal distribution (6.76) with parameters $\mu_{1}, \mu_{2}, \sigma_{1}, \sigma_{2}, \rho$.

The joint moment generating function of two random variables $X$ and $Y$ is defined to be the function $M(s, t)$ of two real variables defined by
$$
M(s, t)=\mathbb{E}\left(e^{s X+t Y}\right)
$$
for all values of $s$ and $t$ for which this expectation exists. Show that the joint moment generating function of a pair of random variables having the standard bivariate normal distribution (6.73) is
$$
M(s, t)=\exp \left[\frac{1}{2}\left(s^{2}+2 \rho s t+t^{2}\right)\right]
$$
Deduce the joint moment generating function of a pair of random variables having the bivariate normal distribution (6.76) with parameters $\mu_{1}, \mu_{2}, \sigma_{1}, \sigma_{2}, \rho$.

Key Concepts

Bivariate Normal Distribution

The bivariate normal distribution is a two-dimensional extension of the normal distribution, characterized by its mean vector and covariance matrix. It describes the joint behavior of two normally distributed random variables, which may be correlated. A key feature of this distribution is that any linear combination of the variables is also normally distributed, and its joint moment generating function can be expressed in an exponential quadratic form.

Parameter Transformation in Normal Distributions

For normally distributed random variables, especially in moving from the standard (zero mean and unit variance) to the general form with arbitrary means and variances, parameter transformation techniques are used. This process involves standardizing the variables and then applying the properties of the moment generating function to incorporate the effects of shifts (means), scaling (variances), and correlation. This transformation is fundamental to extending results derived for the standard case to more general settings.

Joint Moment Generating Function

The joint moment generating function (MGF) of a pair of random variables is a function that uniquely characterizes their joint distribution by encoding all joint moments. It is defined as the expected value of the exponential function of a linear combination of the random variables. This tool is essential in probability theory and statistics for finding moments, proving convergence, and facilitating transformations of random variables.

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