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 21h comment Consider 2 Stocks. If Stock 1 sells \$10(0.8) or sells \$20(0.9). If Stock 2 sells \$10(0.9) or \$25(0.8). Which stock sells for higher price? You should try instead to solve the following: One given stock always sells for x or y. If it is selling for x today, there are p chances that it will sell for y tomorrow. If it is selling for y today, there are q chances that it will sell for x tomorrow. On the average, which price this stock will sell for? The answer will depend on (x,y,p,q). Then apply this result to stock 1 and to stock 2 and compare the average prices each stock sells for. 21h comment What's the difference between a random variable and a measurable function? These are the same. Searching for a difference, one could note that every measurable function on a measurable space becomes a random variable as soon as one fixes a probability measure on the measurable space to give it the structure of a probability space. Thus, random variables would be measurable functions defined on a probability space, or, to summarize: random variable = measurable function + probability measure. 21h comment Identifying markov chains and the markov property The fact that you think this is all very well but the site asks to explain reasons why you have the proposition of answer you have. 21h comment Marginal distribution for two continous random variables One option, explained several times on the site, is to write rigorously the PDF of $(X,Y)$ from the start, here, f_{X,Y}(x,y)=a\mathbf 1_{x^2