Verify that p(x) = 6x-1 is a probability density function on [1, ∞) and calculate its mean value. %3D
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![Problem 4. Suppose a line segment is divided into two pieces, the longer of length a and the shorter of length b.
a + b
If
a
then this ratio is known as the golden ratio or golden mean, denoted 1.618034. (It doesn't
b.
=
a
a +b
a
matter what a and b are; as long as
the ratio will equal .) See the figure below:
a
a
a+b
1
(a) Show that is a solution to the equation x2 – x – 1 = 0, and find an exact (non-decimal) expression for 4.
a +b
(Hint: In the equation
a
let b = 1 and a = x, so
b'
= x = 4.)
a
(b) Consider the sequence a, defined in the following way:
1
a2 = 1+
1
1
1
a4 = 1+
1+
az = 1+
1+
a1 = 1+1,
az = 1+
1
1+
1+1
1+1
1
1+
1+1
1
1+
1
1 +
1+1
This sequence has a limit (you do not need to prove this), given by
1
L = 1+
1
1+
1
1 +
1+
1+..
What is this limit? (Hint: First explain why L = 1+
.)
Problem 5. Show that lim Vn! = o0. (Hint: Verify that n! > (n/2)"/2 by observing that half of the factors are
greater than or equal to n/2. You may want to consider the cases where n is even andn is odd separately to verify
that this hint is true.)
Vn!
Problem 6. Let b, =
1
(a) Show that In b, = -
k
In -.
k=1
(b) Show that In b, converges to
In x dx. (Hint: Consider a Riemann sum.)
(c) What is the limit of b,? Justify your answer.
1
Problem 7. Let cn =
1
+ -=
2n
1
n+1
-n + k
k=0
n
(a) Calculate c1, c2, C3, and c4.
1
(Hint: Begin by approximating this integral using two
2n
(b) Show that
dx +
< Cn S
dx +
different Riemann sums of f (x) = x-1 over [n, 2n] with Ar = 1, one Riemann sum using left endpoints, and
the other using right endpoints.)
(c) Find lim cn using the squeeze theorem.
n00
Problem 8. Let a, = H, - In n, where H,, is the nth harmonic number, defined by
n
1
1
+...+ - =
+
3
4
1
Hn = 1+
2
k=1
n+1
dx
(a) Show that an >0 for all n > 1. (Hint: Use a Riemann sum to show that Hn >
(b) Show that an is decreasing. (Hint: Consider an
an+1 to be an area.)
(c) Show that lim an exists (you do not need to find this limit).
n00
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