National Taiwan University - Calculus 2 for Class 10-13 (Year 113)
Worksheet 3: Applications in Probability
Name:
ID:
Department:
Introduction.
• Given a continuous random variable X whose domain is the whole real line, a probability density
function
Z ∞(p.d.f.) of X is a non-negative function f (x) defined on R such that
(i)
f (x) dx = 1,
−∞
Z b
(ii) For any interval [a, b], the probability of X ∈ [a, b] is given by P(X ∈ [a, b]) =
f (x) dx.
a
• Given a probability density function f (x) of a continuous random variable X, we define its expected
value E(X), its variance var(X) and its standard deviation σ(X) by
Z ∞
Z ∞
E(X) =
xf (x) dx ;
var(X) =
(x − E(X))2 f (x) dx ;
−∞
σ(X) =
p
var(X).
−∞
In particular, these definitions are defined by improper integrals.
Question 1. Many random variables X are modelled by a normal distribution whose probability density function equals
(x−µ)2
1
e− 2σ2 , where µ, σ are constants and σ > 0.
2πσ
Z ∞
Z ∞
(x−µ)2
√
1
−x2
√
e− 2σ2 dx = 1.
(a) It is known that
e
dx = π. By a suitable substitution, deduce that
2πσ
−∞
−∞
(b) Compute the expected value and standard deviation of X.
(x−µ)2
x2
x
1
(c) Let f (x) = ce− 8 + 4 where c is a constant. Complete the square and rewrite f (x) in the form of √
e− 2σ2 .
2πσ
Z
fX (x) = √
∞
Find c so that f (x) is a probability density density function (i.e.
f (x) dx = 1) and find its expected
−∞
value and standard deviation by just reading the simplified form of f (x).
1
Distribution Function.
For a random variable X with p.d.f. f (x), we define its distribution function F (x) by
Z x
F (x) = P(X ≤ x) =
f (t) dt.
−∞
Using the Fundamental Theorem of Calculus (F.T.C.), we have that F ′ (x) = f (x). This provides us a way to
recover the probability density function from its distribution function.
For example, if we define a new random variableY = aX + b where a > 0, then its distribution function is
Z y−b
a
y−b
F (y) = P(Y ≤ y) = P(aX + b ≤ y) = P X ≤
=
f (x)dx
a
−∞
and hence the p.d.f of Y is given by
F ′ (y) =
d
dy
Z y−b
a
f (x)dx
−∞
F.T.C. 1
=
a
f
y−b
a
.
Question 2. Suppose that a random variable X has probability density function
(x−µ)2
1
fX (x) = √
e− 2σ2 , where µ, σ are constants and σ > 0.
2πσ
Define Y = aX + b where a, b are constants and a > 0.
(a) Write down the distribution function of Y , F (y) = P(Y ≤ y), as an integration of fX (x).
(b) Apply the Fundamental Theorem of Calculus and find the probability density function of Y . What is the
expected value and standard deviation of Y ?
(c) Find constants c and d such that the random variable Z = cX + d has expected value 0 and standard
deviation 1. (In this case, Z is called to have the standard normal distribution.)
2
Question 3. A person with IQ 157 is considered as very smart in a population. In this question, we will clarify this
claim in terms of probability.
(a) Indeed IQ is an example of a continuous random variable. Find the following information regarding IQ
online. What is its probability density function? What are the expected value and standard deviation?
(b) Find constants a and b such that a IQ + b is the standard normal distribution.
(c) Estimate P(IQ ≥ 157) as follows : firstly find a constant c such that P(IQ ≥ 157) = P(X ≥ c) where X
has the standard normal distribution. Then look up the later probability from a table of standard normal
distribution.
3
Question 4.
Suppose X is a continuous random variable with standard normal distribution and set Y = X 2 .
(a) Write down the distribution function of Y as an integral. (You don’t need to evaluate it.)
(b) Find the probability density function of Y . This probability density function is called the Chi squared
distribution.
(c) Compute the expected value and variance of Y .
4