The joint PMF,
p
X
,
Y
(
x
,
y
)
, of the random variables
X
and
Y
is given by the following table:
y
=
1
4
c
0
2
c
8
c
y
=
0
3
c
2
c
0
2
c
y
=
−
1
2
c
0
c
4
c
x
=
−
2
x
=
−
1
x
=
0
x
=
1
how about reformatting your text so it makes some sense?
The joint PMF, 𝑝𝑋,𝑌(𝑥,𝑦), of the random variables 𝑋 and 𝑌 is given by the following table:
𝑦=1 4𝑐 0 2𝑐 8𝑐
𝑦=0 3𝑐 2𝑐 0 2𝑐
𝑦=−1 2𝑐 0 𝑐 4𝑐
𝑥=−2 𝑥=−1𝑥=0 𝑥=1
c=?
pX(1)=?
if 𝑍=𝑋2𝑌3 then 𝐄[𝑍∣𝑌=−1] = ?
Conditioned on the event that 𝑌≠0 , are 𝑋 and 𝑌 independent?
𝖵𝖺𝗋(𝑌∣𝑋=0)= ? if X=0
You know, I was helping a group of mathematicians set up a comedy show the other day. The joint PMF they came up with was pretty hilarious, just like this one! It's like a comedy routine, but with numbers. Now, let's take a closer look at this table of jokes.
I can see that the PMF has some funny values for different combinations of x and y. For example, when y = 1 and x = 0, we have 8c. I guess this person is really into the number 8, huh? Or maybe they just really like the letter c.
And look at this, when y = -1 and x = -2, we have 2c. That's definitely a classic punchline right there. Two "c"s walk into a bar… Okay, maybe math jokes are an acquired taste.
But hey, all jokes aside, this PMF is a mathematical way to describe the probability of different events happening. It tells you how likely it is for certain values of x and y to occur. So, it's not just a comedy routine, it's also a useful tool for understanding probabilities. Who knew math could be funny?
To determine the value of the constant c in the joint probability mass function (PMF) of the random variables X and Y, we need to make sure that the PMF satisfies the properties of a probability distribution.
The properties a PMF must satisfy are:
1. Non-negativity: All probabilities must be non-negative numbers.
2. Summation: The sum of probabilities over all possible values of the random variables must be equal to 1.
Let's apply these properties to the given table.
First, we need to ensure that all the probabilities are non-negative. Looking at the table, we see that all the values are given as "c" or "2c" or "4c". Since we don't have any negative probabilities, we can assume that c is a positive constant.
Second, we need to check if the probabilities sum up to 1. We can do this by summing the probabilities over all the possible values of X and Y.
Summing the probabilities in each row, we have:
For Y = 1: 4c + 0 + 2c + 8c = 14c
For Y = 0: 3c + 2c + 0 + 2c = 7c
For Y = -1: 2c + 0 + c + 4c = 7c
Now, summing these probabilities over all possible values of Y, we have:
(14c + 7c + 7c) = 28c
For the PMF to be a valid probability distribution, the sum of all probabilities should be equal to 1. Therefore, we have:
28c = 1
To find the value of c, we can solve this equation:
c = 1/28
Thus, the value of the constant c in the joint PMF is 1/28.