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A set of N independent, identically distributed random variables ?X(sub n)), with common probability density function p(x), are ordered into a new set of dependent random variables ?X'(sub n)), each with a different probability density function. From this latter set, the n1-th largest random variable through the n(sub M-1)-th largest random variable are selected. Then, the sum of the remaining N+1-M random variables is computed, giving a total of M dependent random variables. The joint probability density function of these M random variables is derived in a form involving a single Bromwich contour integral in the moment-generating function domain. The integral is most easily numerically evaluated by locating (approximately) the real saddlepoint of the integrand and passing the contour through this point. Very high accuracy in the probability density function evaluation is available by using numerical integration instead of a saddlepoint approximation.

A set of N independent, identically distributed random variables ?X(sub n)), with common probability density function p(x), are ordered into a new set of dependent random variables ?X'(sub n)), each with a different probability density function. From this latter set, the n1-th largest random variable through the n(sub M-1)-th largest random variable are selected. Then, the sum of the remaining N+1-M random variables is computed, giving a total of M dependent random variables. The joint probability density function of these M random variables is derived in a form involving a single Bromwich contour integral in the moment-generating function domain. The integral is most easily numerically evaluated by locating (approximately) the real saddlepoint of the integrand and passing the contour through this point. Very high accuracy in the probability density function evaluation is available by using numerical integration instead of a saddlepoint approximation.

A set of N independent random variables (Xn! with arbitrary probability density functions ?pn(x)! are ordered into a new set of dependent random variables, each with a different probability density function. From this latter set, the M-1 largest random variables are selected. Then, the sum of the remaining N + 1 - M random variables is computed, giving a total of M dependent random variables. Several statistics are computed for these M random variables, including their joint probability density function, a quantity called the combined probability and joint probability density function of particular selections, and their distribution. Most of the results can be expressed as a single Bromwich contour integral in the moment-generating domain. This integral is most easily numerically evaluated by locating (approximately) the real saddlepoint of the integrand and passing the contour through that point. Very high accuracy in the joint probability density function evaluations is available by using numerical integration on this latter contour, instead of a saddlepoint approximation.

Schaum's Outline of Probability, Random Variables and Random Processes 

NAS 1.60:1859 Digitized from IA1177308-07-0068 .

Let w be a Kx1 Gaussian random vector with arbitrary Kx1 mean vector r and KxK covariance matrix R. The general quadratic and linear forms of interest are the M random scalars z(m) = w' P(m) w + p(m)' w +p(m) for m=1 :M, where KxK matrix P(m), Kx1 vector p(m), and scalar q(m) contain arbitrary constants for m=1:M. The joint probability density function (PDF) of Mx1 random vector z=Z(1)... Z(M)' at an arbitrary point in M-dimensional space is desired. An exact expression for the joint moment generating function (MGF) of random vector z is derived. The inability (analytic and numerical) to perform the M-dimensional inverse Laplace transform back to the PDF domain requires use of the saddlepoint approximation (SPA) to obtain useful numerical values for the desired PDF of z. A first-order correction term to the SPA is also employed for more accuracy, which requires fourth-order partial derivatives of the joint cumulate generating function (CGF). Derivation of the fourth-order partial derivatives of the CGF involves some interesting and useful matrix manipulations which are fully developed. Two MATLAB programs for the entire SPA procedure (with correction term) are presented in this report.

We consider the problem of finding the optimal upper bound for the tail probability of a sum of $k$ nonnegative, independent and identically distributed random variables with given mean $x$. For $k=1$ the answer is given by Markov's inequality and for $k=2$ the solution was found by Hoeffding and Shrikhande in 1955. We solve the problem for $k=3$ as well as for general $k$ and $x\leq1/(2k-1)$ by showing that it follows from the fractional version of an extremal graph theory problem of Erd\H{o}s on matchings in hypergraphs.

This paper presents the derivation for per-tier outage probability of a randomly deployed femtocell network over an existing macrocell network. The channel characteristics of macro user and femto user are addressed by considering different propagation modeling for outdoor and indoor links. Location based outage probability analysis and capacity of the system with outage constraints are used to analyze the system performance. To obtain the simplified expressions, approximations of ratios of Rayleigh random variables (RVs), Rayleigh to log normal RVs and their weighted summations, are derived with the verifications using simulations.

This paper presents the derivation for per-tier outage probability of a randomly deployed femtocell network over an existing macrocell network. The channel characteristics of macro user and femto user are addressed by considering different propagation modeling for outdoor and indoor links. Location based outage probability analysis and capacity of the system with outage constraints are used to analyze the system performance. To obtain the simplified expressions, approximations of ratios of Rayleigh random variables (RVs), Rayleigh to log normal RVs and their weighted summations, are derived with the verifications using simulations.

Lectures from the National Programme on Technology Enhanced Learning - Electronics and Communication Engineering - Probability and Random Variables

Lectures from the National Programme on Technology Enhanced Learning - Electronics and Communication Engineering - Probability and Random Variables

Lectures from the National Programme on Technology Enhanced Learning - Electronics and Communication Engineering - Probability and Random Variables

Lectures from the National Programme on Technology Enhanced Learning - Electronics and Communication Engineering - Probability and Random Variables

Lectures from the National Programme on Technology Enhanced Learning - Electronics and Communication Engineering - Probability and Random Variables

Includes index

Includes index

Probability; random variables; multiple random variables; functions of random variables; random processes; analysis and processing of random processes; applications

A practical technique is presented for determining the exact probability density function and cumulative distribution function of a sum of any number of terms involving any combination of products, quotients, and powers of independent random variables with H-function distributions. The H-function is the most general named function, encompassing as special cases most of the other special functions of mathematics and many of the classical statistical distributions. Its unique properties make it a powerful tool for statistical analysis. In particular, the product, quotient, and powers of independent H- function variates are also H-function variates, and the Laplace and Fourier transforms and the derivatives of an H-function are readily-determined H- functions. This dissertation provides background material, including history on H-functions and the algebra of random variables and definition, properties and special cases of the H-function. For determining whether convergence of a general Mellin-Barnes integral or an H-function occurs with left-half-plane versus right-half-plane summation of residues, evaluation guidelines are formally established and applied to the known special cases, the Laplace transform, and the derivatives of the H-function. Then, a new, improved formulation for evaluation of an H-function by summing residues is derived.

Assume that there is given an arbitrary n-dimensional probability distribution F. A recursive construction is found for a sequence of functions x sub 1 = f sub 1 (U sub 1, ..., U sub n), ..., x sub n = f sub n (U sub 1, ..., U sub n) such that if U sub 1, ..., U sub n are independent random variables having uniform distribution over the open interval (0,1), then the joint distribution of the variables x sub 1, ..., x sub n coincides with the distribution F. Since uniform independent random variables can be well simulated by means of a computer, this result allows one to simulate arbitrary n-random variables if their joint probability distribution is known.

We consider integrals on unitary groups $U_d$ of the form $$\int_{U_d}U_{i_1j_1}... U_{i_qj_q}U^*_{j'_{1}i'_{1}} ... U^*_{j'_{q'}i'_{q'}}dU$$ We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows us to rederive an asymptotic expansion as $d\to\infty$. Using this we rederive and strenghthen a result of asymptotic freeness due to Voiculescu. We then study large $d$ asymptotics of matrix model integrals and of the logarithm of Itzykson-Zuber integrals and show that they converge towards a limit when considered as power series. In particular we give an explicit formula for $$\lim_{d\to\infty}\frac{\partial^n}{\partial z^n}d^{-2} \log\int_{U_d} e^{zd Tr (XUYU^*)}dU|_{z=0}$$ assuming that the normalized traces $d^{-1} Tr(X^k)$ and $d^{-1} Tr (Y^k)$ converge in the large $d$ limit. We consider as well a different scaling and relate its asymptotics to Voiculescu's R-transform.

We consider integrals on unitary groups $U_d$ of the form $$\int_{U_d}U_{i_1j_1}... U_{i_qj_q}U^*_{j'_{1}i'_{1}} ... U^*_{j'_{q'}i'_{q'}}dU$$ We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows us to rederive an asymptotic expansion as $d\to\infty$. Using this we rederive and strenghthen a result of asymptotic freeness due to Voiculescu. We then study large $d$ asymptotics of matrix model integrals and of the logarithm of Itzykson-Zuber integrals and show that they converge towards a limit when considered as power series. In particular we give an explicit formula for $$\lim_{d\to\infty}\frac{\partial^n}{\partial z^n}d^{-2} \log\int_{U_d} e^{zd Tr (XUYU^*)}dU|_{z=0}$$ assuming that the normalized traces $d^{-1} Tr(X^k)$ and $d^{-1} Tr (Y^k)$ converge in the large $d$ limit. We consider as well a different scaling and relate its asymptotics to Voiculescu's R-transform.

We consider integrals on unitary groups $U_d$ of the form $$\int_{U_d}U_{i_1j_1}... U_{i_qj_q}U^*_{j'_{1}i'_{1}} ... U^*_{j'_{q'}i'_{q'}}dU$$ We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows us to rederive an asymptotic expansion as $d\to\infty$. Using this we rederive and strenghthen a result of asymptotic freeness due to Voiculescu. We then study large $d$ asymptotics of matrix model integrals and of the logarithm of Itzykson-Zuber integrals and show that they converge towards a limit when considered as power series. In particular we give an explicit formula for $$\lim_{d\to\infty}\frac{\partial^n}{\partial z^n}d^{-2} \log\int_{U_d} e^{zd Tr (XUYU^*)}dU|_{z=0}$$ assuming that the normalized traces $d^{-1} Tr(X^k)$ and $d^{-1} Tr (Y^k)$ converge in the large $d$ limit. We consider as well a different scaling and relate its asymptotics to Voiculescu's R-transform.

We consider integrals on unitary groups $U_d$ of the form $$\int_{U_d}U_{i_1j_1}... U_{i_qj_q}U^*_{j'_{1}i'_{1}} ... U^*_{j'_{q'}i'_{q'}}dU$$ We give an explicit formula in terms of characters of symmetric groups and Schur functions, which allows us to rederive an asymptotic expansion as $d\to\infty$. Using this we rederive and strenghthen a result of asymptotic freeness due to Voiculescu. We then study large $d$ asymptotics of matrix model integrals and of the logarithm of Itzykson-Zuber integrals and show that they converge towards a limit when considered as power series. In particular we give an explicit formula for $$\lim_{d\to\infty}\frac{\partial^n}{\partial z^n}d^{-2} \log\int_{U_d} e^{zd Tr (XUYU^*)}dU|_{z=0}$$ assuming that the normalized traces $d^{-1} Tr(X^k)$ and $d^{-1} Tr (Y^k)$ converge in the large $d$ limit. We consider as well a different scaling and relate its asymptotics to Voiculescu's R-transform.

Transforms the outcome of an experiment into a number. De�nitions: Probability Space: (S, A, P) S - sample space, A - events, P - probability Random variable is a function on S with values in real numbers, X:S_R

Transforms the outcome of an experiment into a number. De�nitions: Probability Space: (S, A, P) S - sample space, A - events, P - probability Random variable is a function on S with values in real numbers, X:S_R

Transforms the outcome of an experiment into a number. De�nitions: Probability Space: (S, A, P) S - sample space, A - events, P - probability Random variable is a function on S with values in real numbers, X:S_R

Transforms the outcome of an experiment into a number. De�nitions: Probability Space: (S, A, P) S - sample space, A - events, P - probability Random variable is a function on S with values in real numbers, X:S_R

Transforms the outcome of an experiment into a number. De�nitions: Probability Space: (S, A, P) S - sample space, A - events, P - probability Random variable is a function on S with values in real numbers, X:S_R