I can give you one data point. I have worked on computer vision (implementing, training, and running convolutional neural nets), and have also applied machine l

I can give you one data point. I have worked on computer vision (implementing, training, and running convolutional neural nets), and have also applied machine learning methods on natural language processing. I have never taken analysis and I do not feel like I am missing out. In most applied machine learning fields nowadays, the "math" of the it is used more as a gateway into intuition, not really anything rigorous. For example people in machine learning research nowadays use the word "manifold" a lot, but I would wager most do not have a rigorous background in topology. In applied machine learning, the details of your problem domain matters more than understanding the derivation of your tools, this is true even in computer vision, which used to be a lot more rigorous before convolutional neural nets. However, if you are looking to take a rigorous course in probability, I would recommend analysis since it is the foundation of probability, you cannot derive a lot of basic theorems in probability without it. I took a rigorous class in probability once without analysis, and the gaps in knowledge was Clear everyday. If you are determined to tackle theoretical course just as a confidence builder, then I would recommend linear algebra since it has far wider applications. Linear algebra is also a must if you want to understand many optimization tools used in machine learning. As for finance, I have a no direct knowledge here, but I do know when companies like D.E. Shaw recruit here at Penn for quantitative analysts, a strong math background (including analysis) is a must. Whether they use it on a day to day basis is unclear. However, taking one analysis class will not be sufficient for such firms, they are looking for many years of formal schooling and in some cases, publications.