Suppose the joint density function of X and Y is defined as f(x, y)=1/2
What is the marginal density function of Y?
Answers
Answer:
Again, we deviate from the order in the book for this chapter, so the subsec-
tions in this chapter do not correspond to those in the text.
6.1 Joint density functions
Recall that X is continuous if there is a function f(x) (the density) such that
P(X ≤ t) = Z t
−∞
fX(x) dx
We generalize this to two random variables.
Definition 1. Two random variables X and Y are jointly continuous if there
is a function fX,Y (x, y) on R
2
, called the joint probability density function,
such that
P(X ≤ s, Y ≤ t) = Z Z
x≤s,y≤t
fX,Y (x, y) dxdy
The integral is over {(x, y) : x ≤ s, y ≤ t}. We can also write the integral as
P(X ≤ s, Y ≤ t) = Z s
−∞ Z t
−∞
fX,Y (x, y) dy
dx
=
Z t
−∞ Z s
−∞
fX,Y (x, y) dx
dy
In order for a function f(x, y) to be a joint density it must satisfy
f(x, y) ≥ 0
Z ∞
−∞
Z ∞
−∞
f(x, y)dxdy = 1
Just as with one random variable, the joint density function contains all
the information about the underlying probability measure if we only look at
the random variables X and Y . In particular, we can compute the probability
of any event defined in terms of X and Y just using f(x, y).
Here are some events defined in terms of X and Y :
{X ≤ Y }, {X2 +Y
2 ≤ 1}, and {1 ≤ X ≤ 4, Y ≥ 0}. They can all be written
in the form {(X, Y ) ∈ A} for some subset A of R
2
.
Answer:
The correct answer of this question is the random variable X (or Y) is considered by itself .
Step-by-step explanation:
Given - The joint density function of X and Y .
To Find - What is the marginal density function of Y.
The marginal distributions of X and Y are g(x) = y f (x,y) and h(y) = x f (x,y), respectively ( = summation notation). That's probably all you need to know if you're good at math. It explains how to calculate a marginal distribution. When a pair of random variables (X, Y) is studied separately, the density function of random variable X (or Y) is called the marginal density function.
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