In this is article we are going to learn about the terms two dimensional random variable, cumulative distribution function, marginal probability and joint density function.

Two Dimensional Random Variable

Let E be an experiment and S a sample space associated with E. Let X = X(s) and Y = Y(s) be two function each assigning a real number to each outcomes s Є S. We call (X, Y) a two dimensional random variable.

If X₁ = X₁(s), X₂ = X₂(s), ……., X_n = X_n(s) are n functions each assigning a real number to every outcome s Є S, we call (X₁, X₂, … , X_n) as n-dimensional random variable.

Discrete and Continuous Random Variable

Let (X, Y) be two-dimensional discrete random variable with possible values of (X, Y) are finite or countably infinite. That is, the possible values of (X, Y) may be represented as (x_i, y_j), i = 1, 2, …, n, j = 1, 2, …, m. With each possible outcome (x_i, y_j) we associate a number p(x_i, y_j) representing P[X = x_i , Y = y_j ] and satisfying the following conditions:
\[(i)p({{x}_{i}},{{y}_{j}})\ge 0~~\forall x,y\]
\[(ii)\sum\limits_{j=1}^{\infty }{\sum\limits_{i=1}^{\infty }{p({{x}_{i}},{{y}_{j}})=1}}\]
The function p defined for all (x_i, y_j) in the range space (X, Y) is called the probability function of (X, Y). The set of triplets (x_i, y_j;p(x_i, y_j)) i, j = 1, 2, … is called the probability distribution of (X, Y).

Joint Density Function

Let (X, Y) be a continuous random variable assuming all values in some region R of the Euclidian plane. The joint probability density function f(x, y) is a function, satisfying the following conditions:
\[(i)f(x,y)\ge 0~~\forall x,y\]
\[(ii)\iint\limits_{R}{f(x,y)dxdy=1}\]

Cumulative Distribution Function

Let (X, Y) be a two-dimensional random variable. The cumulative distribution function (cdf) F of the two-dimensional random variable (X, Y) is defined by F(x, y) = P[X ≤ x, Y ≤ y]

Marginal and Condition Probability Distribution

With each two dimensional random variable (X, Y) we associate two one dimensional random variable, namely X and Y, individually. That is we may be interested in the probability distribution of X or the probability distribution of Y.

• Let (X, Y) be a discrete random variable with probability distribution p(x_i, y_j), i, j = 1, 2, … The marginal probability distribution X is defined as \[p({{x}_{i}})=P\left[ X={{x}_{i}} \right]=\sum\limits_{j=1}^{\infty }{p({{x}_{i}},{{y}_{j}})}\] Similarly, the marginal probability distribution of Y is defined as \[p({{y}_{j}})=P\left[ Y={{y}_{j}} \right]=\sum\limits_{i=1}^{\infty }{p({{x}_{i}},{{y}_{j}})}\]
• Let (X, Y) be a two-dimensional discrete random variable with joint pdf f(x, y). The marginal probability density function of X. can be defined as \[g(x)=\int\limits_{-\infty }^{\infty }{f(x,y)dy}\]. The marginal probability density function of Y can be defined as \[h(y)=\int\limits_{-\infty }^{\infty }{f(x,y)dx}\]

Note:
\[P(c\le X\le d)=P(c\le X\le d,-\infty \le Y\le \infty )\]
\[\Rightarrow P(c\le X\le d)=\int\limits_{c}^{d}{\int\limits_{-\infty }^{\infty }{f(x,y)dydx}}\]
\[\therefore P(c\le X\le d)=\int\limits_{c}^{d}{g(x)dx}\]
\[\text{Similarly},~~P(a\le Y\le b)=\int\limits_{a}^{b}{h(y)dy}\]
Definition

Let (X, Y) be a discrete two dimensional random variable with probability distribution p(x_i, y_j). Let p(x_i) and q(y_j) be the marginal pdfs of X and Y, respectively.

The conditional pdf of X for given Y = y_j is defined by
\[p\left( {{x}_{i}}|{{y}_{j}} \right)=P\left[ X={{x}_{i}}|Y={{y}_{j}} \right]=\frac{p({{x}_{i}},{{y}_{j}})}{q({{y}_{j}})}if~q({{y}_{j}})>0\]
Similarly, the conditional pdf of Y for given X = x_i is defined as
\[q\left( {{y}_{j}}|{{x}_{i}} \right)=P\left[ Y={{y}_{j}}|X={{x}_{i}} \right]=\frac{p({{x}_{i}},{{y}_{j}})}{p({{x}_{i}})}if~p({{x}_{i}})>0\]

Definition

Let (X, Y) be a continuous two dimensional random variable with joint pdf ‘f’. Let g and h be the marginal pdfs of X and Y respectively.

The conditional pdf of X for given Y = y is defined by
\[g(x|y)=\frac{f(x,y)}{h(y)},h(y)>0\]
The conditional pdf of Y for given X = x is defined by
\[h(y|x)=\frac{f(x,y)}{g(x)},g(x)>0\]

Independent Random Variable

Let (X, Y) be a two dimensional discrete random variable. We say that X and Y are independent random variables if and only if P(x_i, y_j) = p(x_i)q(y_j) for all i and j. That is P(X = x_i , Y = y_j ) = P(X = x_i )P(Y = y_j)for all i and j, i.e., p(x_i, y_j) = p(x_i)q(y_j) Ɐi,j

Let (X, Y) be a two dimensional continuous random variable. We say that X and Y are independent random variables if and only if f(x, y) = g(x)h(y) for all (x, y), where f is the joint pdf and g and h are the marginal pdfs of X and Y, respectively.

Find the constant k so that,

is a joint probability density function. Are X and Y independent?

Solution:

We observe that f(x, y) ≥ 0 for all x, y if k ≥ 0

Further,
\[\int\limits_{-\infty }^{\infty }{\int\limits_{-\infty }^{\infty }{f(x,y)dxdy}}=\int\limits_{y=0}^{\infty }{\int\limits_{x=0}^{1}{f(x,y)dxdy}}\]
\[=k\left\{ \int\limits_{0}^{1}{\left( x+1 \right)dx} \right\}\left\{ \int\limits_{0}^{\infty }{{{e}^{-y}}dy} \right\}\]
\[=k{{\left[ \frac{{{x}^{2}}}{2}+x \right]}_{0}}^{1}{{\left[ -{{e}^{-y}} \right]}_{0}}^{\infty }=\frac{3}{2}k\]
Accordingly, f(x, y) is a joint probability density function if k=2/3. With k=2/3, we find that the marginal density functions are
\[g(x)=\int\limits_{-\infty }^{\infty }{f(x,y)dy}=\frac{2}{3}\left( x+1 \right)\int\limits_{0}^{\infty }{{{e}^{-y}}dy}\]
\[\therefore g(x)=\frac{2}{3}\left( x+1 \right),0<x<1\]
\[\text{and}~h(y)=\int\limits_{-\infty }^{\infty }{f(x,y)dy}=\frac{2}{3}{{e}^{-y}}\int\limits_{0}^{1}{\left( x+1 \right)dx}\]
\[\therefore h(y)=\frac{2}{3}{{e}^{-y}}.\frac{3}{2}={{e}^{-y}},y>0\]
We observe that g(x)h(y) = f(x, y)
Therefore, X and Y are independent random variable.

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