Vízügyi Közlemények, 1971 (53. évfolyam)

4. füzet - Rövidebb közlemények és beszámolók

(Г,

5) of the best is 97.84%. When using the exponential distribution the lowest probability of fitting is 1.10%, while that of best is 86.33%. In the course of further processing the average probability of fitting has been determined for both types of function. For the three-parameter gamma and the exponential distributions the values obtained were 71.9 and 54.09%, respectively. I "'rom the foregoing it appears safe to conclude that in the movement of bed load Ihe distances covered in individual jumps can be described by the exponential and gamma distribution alike, although the exponential distribution is in closcr agreement with Ihe results of observations. In fact, in the case of the exponential distribution the average of fitting probabilities is close to 50%. In sub-section b) an analogy was considered between the process of decay of radioactive materials and that characterizing the movement of bed-load. It has been verified mathematically that both processes can be described by starting from Eq. (12). The representative data provided by the measurement results (samples) and the results of sample analysis are shown in Table VI. The sample elements (the number n of jumps during the time interval T) may be regarded with great probability as independent random variables originating from the same distribution (population). On the strength of the measurement data (samples

) it has been verified that the probability distribution of the event that a particle performs exactly n jumps during the time interval T is a Poisson — type distribution. As revealed by the data compiled in Table VII the probability of the poorest fit is 9.00%, while that of the best is 65,26%. In sub-section c) the length r

i

n of the distance travelled during the time interval 7', and in determining the distribution function of the virtual travelling velocity, first the case A are discussed, in which the particles perform exactly n jumps during the time interval T. Subsequently the general case (B) is considered, where the number of jumps performed during the time interval T is a random variable. Case A a

) The random variables f

2, ..., follow a gamma distribution. In this case Ihe density function of Ihe random variable y

n is given by Eq. (13), while Ihe distribution function of the virtual travelling velocity by Eq. (16). b

) The random variables £

2, • • •> follow an exponential distribution. In this case the distribution function of the random variable y

n is given by Eq. (20), while the distribution function of the virtual travelling velocity by Eq. (21). On the basis of laboratory measurements it seems safe to conclude that if the probability distribution of the distances covered by a single jump is exponential, then the distribution of the distance y

n travelled in n jumps can be approximated by a gamma distribution (Fig. 9 and Table VII). Case It a) The random variables £„ Í,, .. ., Ç

n which arc not interdependent and do not depend on v(T), follow а ЛР/

( -parameter gamma distribution, while the distribution of è(T) is of the Poisson type. In this case the distribution function of the random variable r

m is given by Eq. (31), while the distribution function of the virtual travelling velocity by Eq. (32). b) The random variables |

2, ..., f

n which are not interdependent and do not depend on v(T) follow an exponential distribution. In tliis case the distribution function of the random variable r/

n is given by Eq. (34), while the distribution function of the virtual travelling velocity by Eq. (35). The distribution function characterizing the virtual travelling velocity of bed load has been determined concretely for Ihe cases of gamma- and exponential type distribution functions. With Lhe help of the general relationships given by Eq. (5) the distribution functions of the virtual travelling velocity can be determined — in a relatively simple manner — even in the cases, where the distances covered by the particles in individual jumps follow a distribution which is variable in type from jump to jump.