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Random Variables : Continuous R.V., Exponenetial, R.V, Mean and Variance

3) Let X be a continuous random variable with probability density function
f(s)= c(1 + s^2) for -2 <= s <= 2.
a) Determine c
b) Determine Pr {X <= 0}
c) Determine the mean of X
d) Why is the previous answer fairly obvious?
e) Determine the variance of X
f) Compute Pr {X = 2 | X = 0}
g) Determine the cumulative distribution function

4) Let Y be an exponential random variable with parameter 1.5
a) Compute Pr {Y=0}
b) Compute Pr {Y > 2}
c) Compute Pr {Y > 4 | Y > 2}
d) Compute Pr {Y > 5 | Y > 3}
e) Compute Pr {Y > 5.6 | Y > 3.6}
f) Compute Pr {Y > x + 2 | Y > x } for x >= 0?
g) What is E[Y]?
h) What is the variance of Y

Solution Preview

This is a pretty big assignment. I'll give you a method for solving each part, and leave the number crunching up to you.

a)Recall that if you integrate the probability density function over the whole domain, you have to get 1. In this case the domain of f(s) is [-2, 2], so integrate f(s)ds from -2 to 2, let the result equal 1, and solve for the constant c. (I get c = 1/(4+16/3) which you can ...

Solution Summary

A cumulative distribution functions is calculated from a random variables and probability density function. Variance is calculated from an exponenetial random variable.