X ∼ exp(λ). Determine PDF of Y for the following cases. Make
sure that you specify...
X ∼ exp(λ). Determine PDF of Y for the following cases. Make
sure that you specify the range for Y values. Be extra cautious
with many-to-one mappings.
(a) Y = X − X2
(b) Y = 16 − X4
(b) Y = cos(X).
Let X ~ exp(λ)
MGF of X = λ/(1-t)
a) What is MGF of Y = 3X
b) Y has a common distribution, what is the pdf of Y?
c) Let X1,X2,....Xk be independent and identically distributed
with Xi ~ exp(λ) and S = Σ Xi (with i = 1 below
the summation symbol, and k is on top of the summation symbol).
What is the MGF of S?
d) S has a common distribution. What is the pdf of...
Find the (probability generating function) p.g.f.’s of the
following distributions:[3+3=6]
•P(X=x) =(exp(−λ)λ^x)/((1−exp(−λ))x!) , for x= 1,2,3,..., and λ
>0.
•P(X=x) =((pq)^x)(1−q^(N+1))^−1,for x= 0,1,..., N; where 0 <
p < 1, p+q= 1.
Let X and Y have joint PDF
f(x) = c(e^-(x/λ + y/μ)) 0 < x < infinity and 0 < y
< infinity
with parameters λ > 0 and μ > 0
a) Find c such that this is a PDF.
b) Show that X and Y are Independent
c) What is P(1 < X < 2, 0 < Y < 5) ? Leave in
exponential form
d) Find the marginal distribution of Y, f(y)
e) Find E(Y)
Consider the ODE y''(x) = λy(x) for some real constant λ.
Determine ALL values of λ for which there exists solutions
satisfying the boundary conditions y(0) = y(10) = 0. For each such
λ, give all possible solutions. Are they unique?
Given Fxy(x,y)=u(x)u(y)[1-Exp(-x/2)-Exp(-y/2)+Exp(-(x+y)/2)].
Note: The step functions mean that Fxy(x,y)=0 for either or both
x<0 and y<0. Any x or y argument range below zero must be
truncated at zero. Determine:
a) P{X<=1,Y<=2} Ans: 0.2487
b) P{0.5<X<1.5} Ans: 0.3064
c) P{-1.5<X<=2, 1<Y<=3} Ans: 0.2423
Let X be a random variable. Suppose X ∼ Exp(0.5). The area under
the pdf of the Exp(0.5) distribution above the interval [0, 2] is
0.6321. What is the value of the cumulative distribution function
FX of X at x = 2?
Provide a diagnosis for the following cases. Make sure
you say why you chose that diagnosis by noting the specific
behaviors described in the case.
The Case of Matthew
Boudreaux Matthew,
a 25-year-old single man, was admitted to a large metropolitan
hospital’s mental health service. He was not verbal and offered no
chief complaint on admission. Matt’s older cousin, Lonny Pasquale,
brought him to the hospital and offered that “Matt is going crazy
again. He probably needs to go back to...
The following rate data was collected for a reaction X + Y ->
Products
Exp.
[X]
[Y]
Rate of formation of a product
1
0.23
M
0.17
M
0.33
M/h
2
0.46
M
0.17
M
0.66 M/h
3
0.23
M
0.51
M
0.99 M/h
a. Determine the orders with respect to the two reactants and
write the rate law for the reaction
b. Calculate the value...
For which values of λ does the system of equations
(λ − 2)x + y = 0
x + (λ − 2)y = 0
have nontrivial solutions? (That is, solutions other than x = y
= 0.) For each such λ find a nontrivial solution.
The joint PDF of X and Y is given by f(x, y) = C, (0<
x<y<1).
a) Determine the value of C
b) Determine the marginal distribution of X and compute E(X) and
Var(X)
c) Determine the marginal distribution of Y and compute E(Y) and
Var(Y)
d) Compute the correlation coefficient between X and Y