Conditional distributions

The probability distribution of $Y$ when $X$ is known to be a specific value is the conditional distribution of $Y$ given $X$, if $X$ and $Y$ are two jointly distributed random variables. Conditional distribution is primarily divided into two classes, as mentioned below:

Conditional discrete distributions

Assume $X$ and $Y$ are two discrete random variables with probability mass function:

We can determine the conditional probability in terms of discrete random variables if we know the value of $Y$:

This provides us with the conditional probability mass function of $X$ given $Y=y$. This probability can be calculated for any possible $x$ value.

Example

Suppose $X$ and $Y$ have joint probability function as:

Compute $P(Y = 0| X = 4)$.

Let's put the values in the formula mentioned above:

Conditional continuous distributions

If $Y$ is a continuous random variable, the probability associated with any particular value of a continuous distribution is null $P(Y = y) = 0$. Therefore, we cannot divide by this. Hence, the probability for conditional continuous functions can be calculated by integration. Let's assume that $h$ is extremely small, and by integrating, we get:

Likewise, when $X ∈ A$ that is any event, we can assert that:

Therefore, for a very small $h$ we have:

Because $h$ is so small, we consider this to be the conditional probability for the continuous random variable $P(X ∈ A | Y = y)$.

Let's evaluate the steps above:

The probability density function for $X$ conditioned on $Y = y$ can be considered to be the integrand.

Example

Joint density functions of $X$ and $Y$ are given as:

Where,

Compute $P(0.3 ≤ Y ≤ 0.4 | X = 0.8)$.

Using the definition given:

Putting values in $4x^2 y + 2y^5$ for $X$ and $Y$. For conditioned $X$ we use $2x^2 + 1/3$ function.

Now we'll calculate the values below:

Relation to independence

Two events $A$ and $B$, are said to be independent if $P(A ∩ B) =P(A) P(B)$. We have an equivalent definition of independence for the random variables $X$ and $Y$as stated below. Intuitively, $X$ and $Y$ independence indicates that $X$ and $Y$ do not influence each other. It means that the values of $X$ do not affect the probability for $Y$ and vice versa.

• If $X$ and $Y$ are two discrete random variables, then $X$ and $Y$ are said to be independent if and only if $p_{Y∣X}(y∣x)=p_Y(y)$, for every $x,y∈R$.

• If $X$ and $Y$ are joint absolute continuous random variables, then $X$ and $Y$ are independent if $f_{Y|X} (y | x) = f_Y (y)$, for every $x, y ∈ R$.

Applications

Conditional distributions are helpful when we collect data for two variables, such as gender and income preference. Still, we are interested in solving probability questions when we know the value of one of the variables. In real life, there are numerous cases where we know the value of one variable and may use a conditional distribution to determine the likelihood of another variable taking on the specific value.