Now, the question here is about the two variables. And if you've been given that what the variables are in, end some, right, that since both video burns are independent So therefore the joint probability mass function can be given by we can see it does be off Exco ma vie There's equal dough your eggs in tow be off by Oh, So, uh, now, based on this formula, I am actually going to draw the table for the joint. Pdf has birth. The first question that is asked of us. So I'm going to first take a table here. So let me just resize this. Yeah, so we've taken the table here and now, to start with the simplification part, you can see the values for X from 0123 You have Britain in the columns, and the horizontal values are alternating from 0 to 4 and they've given us a pmf off Ryan X. And the questions are now using this. Remember, the mass functions were supposed to calculate this joint probabilities. So you got here the first value. As for the formula here, gravel, it probability of X equals zero y called zero will be equal do individual multiplication of probabilities so x equal to zero. The provided Lee here is 0.1 okay. And, uh, indoor for Why? Equal to zero? The property's a 0.1 again. So I do 0.1 and a 0.1. So this gives me the value here as your point 01 So in a similar way, Amado, multiply The terms are probably clean and formula this country table. So this gives me 0.2 Then the next I have is your point 03 Then I have a zero point zero again. I have 0.0 do similarly in completing the other value on So this is 0.3 Then I have 0.6 The next is you don't 0.9 Then again, if you don't mind, you know six Then I have 0.6 for the told road. Next is a 0.0 full and then this will be 0.0 Cool. Found out we have 0.8 This is 0.4 The next is 0.1 do. Then this will be 0.6 Then I have a 0.0 eat and this is 0.4 last Childress, 0.8 and 0.4 So you're formally to the people here for the joint off gifts and this values represent the X values. And here's the values of survival was the water could want. So now moving on to the part before the fastest probability off excess lesson equals one, and why is also less any one? So if you see when they say probability of excellence and equal to one, it means it to the value of X equal to zero and X is equal to one and similarly, for why all so we can take X equals y equal to zero and viacord one. So we have those probabilities. You're the four probabilities. You can see these other four ones one toe, three and four. Someone would take these four bodies and since it represents, or so our stands for additions and we'll do add those values So this is going to be P off X equal to zero less be off. X is equal to one. Let's be off. Why good resume and the lastest place for your foreign corridor One. So we add the probabilities you have 0.1 plus, this is your point. You two extra 0.3 and 0.6 So this value is 0.1 toe. Next the board to ask probability off ex less and equaled one savages Viability off X equal to zero. Less probability of X is equal to one. So from the table, you can see the values are from the BMS, so x equal to zero A 0.1 and X is equal to one is zero point to this will be 0.1 less zero point do this gives me 0.3 since only x the individual valuable probabilities and also be ableto values from the probably mass function values not the joint driving's next 40 of us just provided your violence and he pulled one. So which is again probability of I equals zero plus probability of my ankle, the one so I'm taking these values from here the PM for fight which will be 10.1 less 0.3. So this gives me point for now. You're supposed to take it from multiplication Terram is applicable or this particular statement here. So now if I multiply, be off X There's any good one in Toby off. Why? Listen, Equaled one. So which is 0.3 in tow? 0.4. So this gives me 0.1 so you can see or hear it. If this is the question one and this is a question toe, we can see that photo. Both the term's over here are equal, so I can therefore conclude your that be off the joint provided the ex less and equaled one comma while is unequalled, one can build an aspirin. Gravity off ex has an equal one in two probability off. Why there's an equal do one. So we're done with the party now moving on to the next part that this part see for the park. See, they told us to find the value off Buddha valuables such that it is less than equal to one. So here are able to consider the values of X sent by such that the addition off them will be less than equal one. So then the possibility that I can take your into consideration is Sebi off. We have exes you and why it is you. Then I can take when p off x zero and wise one because the edition comes as one. And, uh, next weekend, Vegas even X is one. And why is zero apart from these? We contact the other probabilities because of some off these values has to be less than required one. So now I'm going to add their joint formalities from the deep. And so this is 0.1 less 0.3 less little points. You don't do so If you add them, begin the value as your 0.6 clued in with the bar to see going on to the next part question the it says performing the probabilities then X is equal to zero mint. Why be equal to one or two or three or we're going to find the probability off the X is equal. Do 01 tow 34 Mumbai is equal to zero. So start with this part. I'm willing to consider the addition of the joint probabilities where the possibly result that probably suppose I take X as our zero. And why s one symbol a pair like this X equal to zero with 123 values off. Why? So it gives me be off. Then I have zero and to then we have a body off zero with Cleveland X zero advise Wonder Tree. The next is off again. So we add. And now we have excessive on via zero. So this is 00 Then we have be off excess one. And why's zero then? I have be off X is to y zero. Next, I have be off excess three by a zero and the lastest be off excess for advise zero. So taking these values from the table, we can get the answer You're so I'm going to add them. We have 0.3 plus 0.40 point. You don't know. Next is 0.1 Bless wines. You don't do class points in a tree and points it will do again Then I have warned you will too once again. So if you see then the addition off all these values is 0.19 So we're done with all the four parts of this particular