Sum of Two Random Variables

Hint this is most easily done by projecting each distribution into one dimension, where it becomes a Gaussian, and using the theorem from statistics that the variance of the difference (or sum) of two random variables is the sum of the individual variances. 

What happens if two random variables are added together (Figure A.l) Mean the mean of the sum of two random variables is... 

A third method to simulate random variables is convolution, where the desired random variates are expressed as a sum of other random variables that can easily be simulated. For example, the Erlang distribution is a special case of the Gamma distribution when the shape parameter is an integer. In this case, an Erlang random variate with shape parameter can be generated as the sum of j3 exponential random variates each with mean a. A last method to simulate random variables is decomposition (sometimes called composition), where a distribution that can be sampled from is composed or decomposed by adding or subtracting random draws into a distribution that cannot be simulated. Few distributions are simulated in this manner, however. These last two methods are often used when the first two methods cannot be used, such as if the inverse transformation does not exist. 

Variance of the sum of random variables. One can write for any choice of two random variables that... 

On the other hand, Eq. (3-233) states that A is the sum of two statistically independent, Poisson distributed random variables Ax and Aa with parameters n(t2 — tj) and n tx — t2) respectively. Consequently,49 A must be Poisson distributed with parameter n(t2 — tx) + n(t3 — t2) = n(t3 — tx) which checks our direct calculation. The fact that the most general consistency condition of the type just considered is also met follows in a similar manner from the properties of sums of independent, Poisson distributed random variables. 

The proof that the variance of the sum of two terms is equal to the sum of the variances of the individual terms is a standard derivation in Statistics, but since most chemists are not familiar with it we present it in the Appendix. Having proven that theorem, and noting that AEs and AEr are independent random variables, they are uncorrelated and we can apply that theorem to show that the variance of AT is ... 

The density function of the sum of two independent continuous random variables is computed by the convolution of the two probability densities. Loosely speaking, two random numbers are independent, if they do not influence each other. Unfortunately, convolutions are obviously important but not convenient to calculate. 

Another way of handling changes of variables is through the moment generating function. If Z is the sum of two independent random variables X and Y, integration of the two variables under the integral can be carried out independently, hence... 

Example 2.10 Expectation of Discrete Random Variable. Suppose two dice are tossed (Example 2.8). Let Xi be the sum of two dice and X2 and X3 be the values of the first and second tossings, respectively. What are the expectations of Xi, X2 + X3 ... 

In some situations (consider adding some noise to a clean speech signal) the distribution of the sum of two independent random variables, x and y, is required Let... 

In the event that the interval between two tests is stochastic, the situation can be modeled as the sum of a random number of random variables, where the interval — Si can be written as ... 

Before presenting alternative tests about dispersion we have to clarify the calculation law for the dispersion of the sum of two or more independent random variables. To this aim we proceed sequentially as following ... 

Another important result states that the characteristic function of a sum of statistically independent random variables is the product of the characteristic functions of the individual summands. The reader should compare this statement with the deceptively similar sounding one made on page 154, and carefully note the difference between the two. The proof of this statement is a simple calculation... 

Equation 41-A3 can be checked by expanding the last term, collecting terms and verifying that all the terms of equation 41-A2 are regenerated. The third term in equation 41-A3 is a quantity called the covariance between A and B. The covariance is a quantity related to the correlation coefficient. Since the differences from the mean are randomly positive and negative, the product of the two differences from their respective means is also randomly positive and negative, and tend to cancel when summed. Therefore, for independent random variables the covariance is zero, since the correlation coefficient is zero for uncorrelated variables. In fact, the mathematical definition of uncorrelated is that this sum-of-cross-products term is zero. Therefore, since A and B are random, uncorrelated variables ... 

Summing up, when the particle environment is a thermal bath, the two fluctuation-dissipation theorems are valid. In both theorems the bath temperature T plays an essential role. In its form (157) or (159) (Einstein relation), the first FDT involves the spectral density of a dynamical variable linked to the particle (namely its velocity), while, in its form (161) (Nyquist formula), the second FDT involves the spectral density of the random force, which is a dynamical variable of the bath. 

Example 3. For the throw of two fair dice (i.e. each number has 1/6 probability of occurring), let jr be a random variable that is equal to the sum of the numbers then... 

Consider a function/(S, f) dependent on two variables S and t, where S follows a random process and varies with t. If S, is a continuous-time process that follows a Wiener process W then it directly influences the function/() through the variable t in/(S t). Over time, we observe new information about W, as weU as the movement in S over each time increment, given by dS,. The sum of both these effects represents the stochastic differential and is given by the stochastic equivalent of the chain rule known as It s lemma. So, for example, if the price of a stock is 30 and an incremental time period later is 30 /i, the differential is Vi. 

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