Continous Random VariablesDate: 03/15/2001 at 00:37:10 From: Brian Subject: Continous random variables Let c be a constant and consider the density function f(y) = (1/c)e^(-y/2) if Y > or = 0 and (1/c)e^(y/2) if Y < 0 1. find the value of c 2. find the cumulative distribution function F(y) 3. compute F(1) 4. compute P(y>.5) Date: 03/18/2001 at 22:54:50 From: Doctor Jordi Subject: Re: Continous random variables Dear Brian, The exercise that you present above should help you understand continuous random variables and their distributions. I will try to give you a few hints that should help you to solve it: 1) The value of c should be such that the total probability integrates to 1. My suggestion is to find the integral of the density function from minus infinity to plus infinity without paying any attention to the value of c for the moment. You will get some number as an answer. Choose c such that this number times (1/c) equals 1. 2) The cumulative distribution function F(y) has the same value as P(Y < y); it is the probability that the random variable Y takes on a value less than y. Given the density function, f(y), F(y) is defined to be Integral[-infinity, y] (f(s)ds) ^ | (the definite integral of f(s)ds from negative infinity to y. I will write this from now on as Int[-inf, y] (f(s)ds) ) where s is just any dummy variable. In this particular problem, you will have to split up your integration into two intervals, since the form of the density function changes at y = 0. You will get something like this: / Int[-inf, y](f(s)ds) for y =< 0 | F(y) = { | Int[-inf, 0](f(s)ds) + Int[0, y](f(s)ds) for y > 0 \ If you want to check your work, here are some properties that F(y) should have in the end. These properties follow because F(y) is a probability function (or a distribution). F(-inf) = 0 F(inf) = 1 0 < F(y) < 1 for all real y. F(y) is a strictly increasing function (meaning F'(y) is always nonnegative) F'(y) = f(y) 3) This one should be easy; since once you've found F(y), all you have to do is evaluate it a point. 4) You only have to recall an important law about probability. P(Y > y) = 1 - P(Y < y). Further, recall that P(Y < y) = F(y). I hope this is enough to get you started on working this problem. Please write back if you have more questions, or if you would like to talk more about this. - Doctor Jordi, The Math Forum http://mathforum.org/dr.math/ |
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