Obtain the characteristic function for the Poisson law XXXXXXXXXXand use it to verify XXXXXXXXXX). For the Uniform[0, 1] distribution, obtain the characteristic function and verify XXXXXXXXXX). For...

Obtain the characteristic function for the Poisson law (1.5.52) and use it to verify (1.5.53).

For the Uniform[0, 1] distribution, obtain the characteristic function and verify (1.5.55).

For the gamma distribution (1.5.56), derive the characteristic function. Hence or otherwise, derive the characteristic function for the chi-squared distribution with k degrees of freedom.