We compute statistical properties of weak gravitational lensing by large-scale structure in three Cold Dark Matter ( CDM ) models : two flat models with ( \Omega _ { 0 } , \lambda _ { 0 } ) = ( 1 , 0 ) and ( 0.3 , 0.7 ) and one open model with ( \Omega _ { 0 } , \lambda _ { 0 } ) = ( 0.3 , 0 ) , where \Omega _ { 0 } and \lambda _ { 0 } are the density parameter and cosmological constant , respectively .
We use a Particle-Particle/Particle-Mesh ( P ^ { 3 } M ) N -body code to simulate the formation and evolution of large-scale structure in the universe .
We perform 1.1 \times 10 ^ { 7 } ray-tracing experiments for each model , by computing the Jacobian matrix along random lines of sight , using the multiple lens-plane algorithm .
From the results of these experiments , we calculate the probability distribution functions of the convergences , shears , and magnifications , and their root-mean-square ( rms ) values .
We find that the rms values of the convergence and shear agree with the predictions of a nonlinear analytical model .
We also find that the probability distribution functions of the magnifications \mu have a peak at values slightly smaller than \mu = 1 , and are strongly skewed toward large magnifications .
In particular , for the high-density ( \Omega _ { 0 } = 1 ) model , a power-law tail appears in the distribution function at large magnifications for sources at redshifts z _ { s } > 2 .
The rms values of the magnifications essentially agree with the nonlinear analytical predictions for sources at low redshift , but exceed these predictions for high redshift sources , once the power-law tail appears .

We study the effect of magnification bias on the luminosity functions of high-redshift quasars , using the calculated probability distribution functions of the magnifications .
We show that the magnification bias is moderate in the absence of the power-law tail in the magnification distribution , but depends strongly on the value of the density parameter \Omega _ { 0 } .
In presence of the power-law tail , the bias becomes considerable , especially at the bright end of the luminosity functions where its logarithmic slope steepens .
We present a specific example which demonstrates that the bias flattens the bright side logarithmic slope of a double power-law luminosity function .