# The function of a random variable

In this article, we discuss the function of a random variable.

The random variable itself is a function and we are interested in the function of the random variable so it's essentially the function of a function.

Let’s say we have a function ‘**x²**’, then we can define a new function on top of it say **cosine(x²)** so here also we have the function of a function.

It’s important in the case of a random variable as the function of a random variable itself is a random variable. The outcome of an experiment is going to be random and since random variable maps the outcome to some values, the output from a random variable is going to be random and hence the value that the function of a random variable can take is going to be random.

Here is an example, let’s say we have a random variable ‘**X**’ which maps the outcomes of rolling two dice to the sum of two numbers

Let’s say we have another random variable ‘**Y**’ which is a function of random variable ‘**X**’

The way random variable ‘**Y**’ works is that if the sum(of the numbers of two dice, output from the random variable ‘**X**’) is less than 5, then it maps to say 1 point if the sum is from 5 to 8, then Y maps to 2 points, and so on.

We are interested in the expected value of this random variable ‘**Y**’

There are two ways to compute the expected value of the random variable ‘**Y**’: one would be to compute the distribution of ‘**Y**’ i.e the probability of each value that the random variable ‘**Y**’ can take and we can then use the standard formula to compute the expected value of ‘**Y**’. The other way would be to use the distribution of the random variable ‘**X**’ and since ‘**Y**’ is a function of ‘**X**’, we leverage the distribution of ‘**X**’ and compute the expected value of ‘**Y**’ using that.

Let’s compute the expected value using the first approach:

It’s the weighted sum of the values that the random variable ‘**Y**’ can take wherein the weights corresponds to the probabilities of corresponding values

**pᵧ(1)** is the probability that the random variable ‘**Y**’ can take on the value 1

Now the random variable ‘**Y**’ can have the value as 1 if the random variable ‘**X**’ can take on values as 2, or 3, or 4 and ‘**X**’ can take on either of these values when we have the corresponding outcomes as { (1,1), (1,2), (2,1), (1,3), (2,2), (3,1) } or we can call it as the union of three disjoint events

And we can compute the probability that the random variable ‘**Y**’ takes on the value as 1 as below:

‘1/36’ is the probability of **A₁**

‘2/36’ is the probability of **A₂**

‘3/36’ is the probability of **A₃**

Similarly, we can compute the probability of the case that the random variable ‘**Y**’ takes on the value 2, in this case, the marked 4 events in the below image are the outcomes of interest

Similarly, we have the probability that the random variable ‘**Y**’ can take the value as 3 as the following:

Let’s compute the same thing using the other approach:

We already know the distribution of ‘**X**’ but we ignore it in the first approach and compute the distribution of ‘**Y**’

We notice that there is a mapping between ‘**X**’ and ‘**Y**’ for example ‘**Y**’ being 1 equals the scenario that **X equals 2 union X equals 3 union X equals 4** and the probability of these are already known in the distribution of ‘**X**’

Because ‘**Y**’ is a function of ‘**X**’, there is already a relationship between the distribution of ‘**Y**’ and the distribution of ‘**X**’.

Below is the expected value of ‘**Y**’ as from the first approach

We can open up the brackets and the above result can be written as:

Now we already got an expression for the expected value of ‘**Y**’ in terms of the probability of random variable ‘**X**’ taking on different values and interesting thing to note is that the probabilities of all the terms/values that the random variable ‘**X**’ can take are included in this formula

We can write this as below:

where the function **g()** corresponds to the value of ‘**Y**’ corresponding to the input to **pₓ()** for example when we are computing **pₓ(2)** we are interested in the probability of the random variable ‘**X**’ taking on the value 2, this quantity is then multiplied by **g(2)** which would be the value of ‘**Y**’ for input/x as 2.

And we can write the formula compactly as below:

It says that to compute the expectation of ‘**Y**’, sum over all the values that the random variable can take, take probabilities of those values of ‘**X**’ and multiply it with the value that the function **g()** would take for the respective value of ‘**X**’.

So, this is a formula for the expectation of the random variable ‘**Y**’ which is a function of the random variable ‘**X**’ in terms of the distribution of ‘**X**’.

And both the approach gives the same result at the end.

References: PadhAI