Choose a number X at random from the set of numbers...

Choose a number $X$ at random from the set of numbers $\{1,2,3,4,5\}$. Now choose a number at random from the subset no larger than $X$, that is, from $\{1, \ldots, X]$. Call this second number $Y$. (a) Find the joint mass function of $X$ and $Y$. (b) Find the conditional mass function of $X$ given that $Y=i$. Do it for $i=1,2,3,4,5 .$ (c) Are $X$ and $Y$ independent? Why?

Choose a number $X$ at random from the set of numbers $\{1,2,3,4,5\}$. Now choose a number at random from the subset no larger than $X$, that is, from $\{1, \ldots, X]$. Call this second number $Y$.
(a) Find the joint mass function of $X$ and $Y$.
(b) Find the conditional mass function of $X$ given that $Y=i$. Do it for $i=1,2,3,4,5 .$
(c) Are $X$ and $Y$ independent? Why?

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Key Concepts

Joint Probability Mass Function

The joint probability mass function (PMF) describes the probability that two discrete random variables simultaneously take on specific values. This concept is central to understanding the relationship between random variables in a multi-dimensional probability space, where the sum of the joint probabilities over all possible pairs equals one.

Conditional Probability Mass Function

The conditional probability mass function explains the likelihood of one random variable assuming a particular value given that another random variable has a specific value. It is derived by dividing the joint probability by the marginal probability of the conditioning event, and it provides insight into how the behavior of one variable is influenced by the known outcome of another.

Independence of Random Variables

Two random variables are considered independent if the occurrence of one does not affect the probability distribution of the other. In terms of the joint PMF, this means that the joint probability equals the product of their marginal probabilities for all pairs of values. Testing for independence helps determine whether the variables functionally influence each other or behave separately.

Transcript

00:03 All right, so we have a set x is 1, 2, 3, 4, 5.

00:10 We are picking a random variable, i mean a number randomly from this number.

00:16 So that is x.

00:18 And we are forming a set 1, so on that x.

00:22 And then we are picking randomly another number, which is random variable y.

00:26 So let's start.

00:28 What is the probability that x is 1? obviously, it's 1 by 5.

00:32 Because when you pick x is 1, the only subset that you can form.

00:36 Is 1 and the probability that you choose 1 is 1.

00:43 How about probably that x is 2 y is 1? in that case it is 1 by 5 into 1 by 2 because the subset will be 1 comma 2.

00:55 The probably that you choose 1 from 1 come 2 is half.

00:58 So 1 by 5 to 1 by 2 is 1 by 10.

01:00 1 by 5 is probably picking 2 from this and this is same probability that x is 2, y is 2.

01:09 So like that we'll get a table.

01:11 So let's form a table nice table.

01:14 So let's form a nice table.

01:16 So let's keep x here, y here.

01:18 X is 1, 2, 3, 4, 5.

01:20 Y is 1, 2, 3, 4, 5.

01:23 What is the probability that x is 1, 1, 1, 1 by 5, and all these will be 0s? what is the probability that when x is 1 is 1 is 1 is 1 by 10 and this is also 1 by 10 and all these will be 0.

01:38 And this will be 1 by 15 because 1 by 5, it 1 by 3.

01:42 This is also 1 by 15, 1 by 15, this is 0 .0...

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