It is not a pre-requisite, and you can learn it easily once you encounter something you are not familiar with. Statistics is pretty old and there are many learning resources on the Web, which you can get to whenever you hit the wall while learning about deep learning.
As to which field is pre-requisite, I think it is enough to first learn about Gaussian (normal) distribution, linear regression, and logistic regression. Then when you encounter something you don't understand, it is time to invest your time on statistics.
The more requisite fields I believe are calculus and linear algebra. If you haven't learned about them (such as partial derivative, matrix transpose, etc), it is very difficult to start to learn deep learning.
Also, the prior exposure to some of machine learning algorithms would make your learning faster. But I'm sure you already got it given that you finished Andrew Ng's course on Coursera. Andrew Ng starts deep learning course on Coursera from August 15th, so you can join it and get a grasp of what is required.
The book you linked sounds more like focused on machine learning than on statistics, BTW.