A sum of $50,000 is invested at a rate R, selected from a uniform distribution on the interval (0.03, 0.07). Once R is selected, the sum is compounded instantaneously for a year so that X= 50000 e^R dollars is the amount at the end of that year.

a) Find the distribution function and the p.d.f. of X.

b) Verify that X = 50000e^R is defined correctly if the compounding is done instantaneously.

Solution Summary

This solution answers questions regarding the distribution of functions with random variables.

Suppose X and Y are jointly continuous with joint pdf f(x,y)=xe^[-x(1+y)], x,y>0.
a) What are the marginal pdf's of X and Y?
b) What is the conditional pdf of X given Y=y?
c) What is the conditional pdf of Y given X=x?
d) What is the distribution of Y given X=x called?

5.3.-1 Let X1 and X2 the independent Poisson randomvariableswith respective means lamda1 = 2 and lamdaT=3.
Find:
(a) P(X1 = 3, X2 = 5)
(b) P(X1 + X2 = 1)
HINT: Note that this event can occur if and only if {X1 = 1, X2 = 0} or {X1 = 0, X2 = 1}
5.3-3. Let X1 and X2 be independent randomvariableswith probability d

See the attached file.
1) Let Y be a random variable with a density function given by
f(y1) = 3/2y^2 , -1 distribution.
b) find the density function of U2 = 3-Y using the method of distribution.
c) find the density functi

I am having trouble with a few characteristic functions, as described in the attachment. Any help would be greatly appreciated.
The attachment contains a definition of what I mean by a characteristic function.
Thank you!

I've struggled for 3 days to come up with something approaching a relevant answer but am now desperate.
Could you solve Q3, both a) and b) parts from the Exercise Sheet attached?
Happy to pay 2 credits for both answers.
Thank you very much.
3. The random variable X has an exponential distributionwith mean ยต. Let Y

Please choose the correct answer and write briefly why.
A property of continuous distributions is that:
a. As with discrete randomvariables, the probability distribution can be approximated by a smooth curve
b. Probabilities for continuous variables can be approximated using discrete randomvariables
c. Unlike discret