Question

Which of the following functions are probability mass functions? For those that are not, find (if possible) a constant a so that a · p(ω) is a probability mass function.

a. p(ω) = ω 2 55 , ω = 1, 2, 3, 4, 5

b. p(ω) = 1 3 2 3 ω , ω = 3, 4, 5, 6, . . .

c. p(ω) = 1 for each ω in a nine-member set Ω.

d. p(ω) = 1 for each ω in a countably infinite set Ω.

e. p(ω) = ω, ω = 1, 2, 3, 4, . . . , N

f. p(ω) = 1 4 ω , ω = 0, 1, 2, 3, 4, . . .

g. p(ω) = 1 ω , ω = 1, 2, 3, 4, . . .

h. p(ω) = 1 3 (ω − 2), ω = 0, 1, 2, 3, 4

Answer #1

**Answer:**

**a)** p(w) = w^2/55 = ( 1+4+9+16+25)/55 = 55/55 =
1 **Yes it is PMF**

**b)** p(w) = (1/3)(2/3)^w =
(1/3)(2/3)^3+(1/3)(2/3)^4+(1/3)(2/3)^5+(1/3)(2/3)^6+........Its a
GP

= (1/3)(2/3)^3 / (1 -2/3)

= 1/3 * 8/27 * 3

=8/27** : Not a PMF , a = 27/8**

c) p(w) =1 Not a pmf , a = 1/9

Explanation: It is actually for nine members : (1+1+1+1+1+1+1+1+1) = 9

Hence a =1/9 is chosen to make it as 1.

d) p(w) =1 Not a pmf , a = 1/N

Explanation: It is actually for countably infinite , so let N be the count : (1+1+1+1+1+1+1+1+1+........+1)----> N times = N

e) p(w) = w

= 1+2+3+4,.......N

= N+1

f) p(w) = (1/4)^w

= from b we can conclude f is also not a PMF

g) p(w) = 1/w

= 1/1+1/2+1/3 + 1/4....

= not a PMF

h) p(w) = 1/3(w-2)

= 1/3(0-2)+ 1/3(1-2)+ 1/3(2-2)+ 1/3(3-2)+ 1/3(4-2)

= -2/3+-1/3+0+2/3

=1/3

For each of the following joint probability mass functions, make
a table, find the two marginal mass functions, and decide whether X
and Y are independent. (In every case the mass function is defined
to be 0 at values of x and y not specified.)
a. p(x, y) = 1 36 if both x and y are in the set {1, 2, 3, 4, 5,
6}.
b. p(x, y) = x 2y 84 for x = 1, 2, 3 and...

Let X be a random variable with probability mass function
P(X =1) =1/2, P(X=2)=1/3, P(X=5)=1/6
(a) Find a function g such that E[g(X)]=1/3 ln(2) + 1/6 ln(5).
You answer should give at least the values g(k) for all possible
values of k of X, but you can also specify g on a larger set if
possible.
(b) Let t be some real number. Find a function g such that
E[g(X)] =1/2 e^t + 2/3 e^(2t) + 5/6 e^(5t)

Which of the following are valid probability distribution
functions? (Hint: The P values have to follow the rules previously
given for probabilities.)
x
P(x)
5
0.6
6
0.8
7
– 0.4
x
P(x)
1
.1
2
.2
3
.3
4
.4
x
0
1
2
3
4
P(x)
1/6
1/6
1/6
1/6
1/6

2. Show that each given function is a probability density
function and find the indicated probabilities:•
Let f(x) =1/4(1+x)^−5/4 in [0,∞);
find
P(0≤X≤2), P(1≤X≤3) and P(X≥5)

Is the following a probability mass function (probability
distribution function)? Show why or why not.
P(X=x) = 3 / (4 *(3-X)!
*X!) for x = 0, 1, 2,
3
and zero elsewhere
What do they mean by, "and zero elsewhere", other than
that I understand the question. Thanks!

Consider the functions f(x) and g(x), for which f(0)=2, g(0)=4,
f′(0)=12 and g′(0)= -2
find h'(0) for the function h(x) = f(x)/g(x)

Determine which of the following functions are injective
(one-to-one) on their respective domains and codomains
(a) f : ℝ → [0,∞), where f(x) = x²
(b) g : ℕ → ℕ, where g(x) = 3x − 2
(c) h : ℤ_7 → ℤ_7, where h(x) ≡ 5x + 2 (mod 7)
(d) p : ℕ ⋃ {0} → ℕ ⋃ {0}, where p(x) = x div 3

Which of the following set of quantum numbers (ordered n, ℓ, mℓ,
ms) are possible for an electron in an atom? Check all that
apply.
A) 5, 3, -3, 1/2
B) 2, 1, 0, 1
C) 4, 3, -2, 1/2
D) 4, 3, -4, -1/2
E) 5, 3, 4, 1/2
F) 4, 2, -1, -1/2
G) -1, 0, 0, -1/2
H) 2, 2, 2, 1/2

For each of the following functions fi(x), (i) verify that they
are legitimate probability density functions (pdfs), and (ii) find
the corresponding cumulative distribution functions (cdfs) Fi(t),
for all t ? R.
f1(x) = |x|, ? 1 ? x ? 1
f2(x) = 4xe ?2x , x > 0
f3(x) = 3e?3x , x > 0
f4(x) = 1 2? ? 4 ? x 2, ? 2 ? x ? 2.

The table below shows values for four differentiable functions.
Suppose we know the following things:
h'(x) = g(x)
g'(x) = f(x)
f'(x) = b(x)
b'(x) = h(x)
0
1
2
3
4
b(x)
2
4
3
1
0
f(x)
1
4
2
3
0
h(x)
2
0
3
1
4
g(x)
2
0
4
3
1
a) What is intergal from a = 2 and b = 4 f(x) dx?
b) What is intergal from a = 0 and b =...

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