How can you change a bad course run by somebody else? I am a (very) junior post doc in a prestigious university, and for the last three years I have taught a first-year undergraduate level course on statistics, designed to be taken by biochemists. By "teach", I mean "provide classes going over a set of problem sheets written by somebody else". My background is in particle physics, and I find that the course, written and lectured by a very senior academic, is exceptionally poor. 

For example, the "standard tables" of numerically integrated distributions that he provides to the students differ to the values one gets in R, or using other modern numerical methods. Additionally, his model solutions include things such as assuming that radioactive decay is normally distributed (it isn't, it's a classic example of a Poisson process), cherry-picking data to fit a regression line to, and not making explicitly clear how the different types of expressions vary for the t-distribution when the samples are assumed to have equal or unequal variance. He treats parametric and non parametric tests as interchangeable in answering questions, and the style of the questions themselves boil down to "guess an expression and put numbers into it". There is no mention of Bayes's theorem anywhere, nor of modern statistics. 

The students do not like the course at all, and I find that I spend a large portion of the time filling in gaps in their knowledge, and saying things like "The way  answers this question is as follows", even if I am relatively certain that in doing so he has made a tacit assumption that might not be correct. Whilst admittedly teaching statistics to biochemists is notoriously difficult, I feel that the course as stands does not prepare them for a research career (they can't pin-the-statistical-test-tail-on-the-donkey-of-experimental-design) and actively avoids adequately discussing several quite important subtle points.

I have made most of these points to the senior academic in charge several times. He does not respond to them. He recently discovered that the notation he uses is inconsistent with that of the rest of the literature on the subject, and wrote an email to the people who deliver his classes saying something along the lines of "I have been teaching this course for 20 years..." and then stating that would essentially ignore this fact. When he went on sabbatical for a year, another academic (less junior than I, but still junior) entirely re-wrote the course, and it received much better reviews. When the lead tutor came back, he ignored the changes, and reverted back to his stock material. 

This is incredibly frustrating. I do not know how adequately change anything, without committing career suicide and going to person in charge over the lead's head. I do not wish to make a fuss, but I really feel that it isn't very professional of me to have to start every class essentially ranting about what the students have been taught, teaching them how to pass the exam, and then telling them "fun things" that I think they should know at the end if there is extra time. I want to teach a course that adequately prepares individuals for their future, not very much for the past. 

What is the best way to go about getting out of this situation without either (a) getting me fired, (b) getting the senior academic fired or retired, or (c) creating a very public fuss and generally being a bother?
I commend you for caring about these issues. When you have a permanent position somewhere, I hope that you will push for the sorts of improvements you described.

For the moment, you are in a position of essentially no power, and it is unlikely that anyone will listen to you. You tried the only potential solution at your disposal: you raised your concern to the academic in charge, and you found he doesn't care.

As such I recommend that you do the best job possible within the limits of your authority, try not to make any enemies, and try not to worry about this much. If a similar situation occurs later in your career, when you are likely to have more clout and authority (and possibly less pressure to publish), you might push harder for change.