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[15 pts:] State the value a that would make each of the following functions a valid probability mass function of a discrete random variable X: f(x) = a X + for x = ...

Question

[15 pts:] State the value a that would make each of the following functions a valid probability mass function of a discrete random variable X: f(x) = a X + for x = 0,1,2,3,4,5 f(x) =a * PA + P?, for x = 0,1,2 f (x) =ax2 _ 1,forx =1,2,3,4

[15 pts:] State the value a that would make each of the following functions a valid probability mass function of a discrete random variable X: f(x) = a X + for x = 0,1,2,3,4,5 f(x) =a * PA + P?, for x = 0,1,2 f (x) =ax2 _ 1,forx =1,2,3,4



Answers

Verify that the following functions are probability mass functions, and determine the requested probabilities. $f(x)=\frac{2 x+1}{25}, \quad x=0,1,2,3,4$ (a) $P(X=4)$ (b) $P(X \leq 1)$ (c) $P(2 \leq X<4)$ (d) $P(X>-10)$

Given function here. If tax two x plus one about 25 we have to verify that submission of effects over all domain Starting from 0 to 4. Should be what Let's verifying. 20 plus one upon 25 less 2 1 plus 1.25 less two plus water apartment define a value for. Mhm Since access belonging to 0 to 4. So the values are Mhm. Finally the value is 25 x 25 which is one. Hence pds themselves. She. Mm. Mhm. Now it is asking the requested probabilities are in the problem Part is the x equals to four. We can clear this a part APX equal to four areas just put the value for so 24 plus 1.25 nine x 25. This or value as calculated above also Be excellence than equal to one will be value zero and 1. So we can see P X less than equal to one as zero plus the evil and the value we can just take from the above mentioned Values. one by 25 plus three were 25 four by 2030 access belonging to 2 to 4. That's good. Have an equal to two in less than four. Then the value will be included. Us x equal to two and X equal to three or will not be needed to include health. These values are from the boat expressiveness 5 25 plus seven by 25. Yeah And the last one is 20 x bigger than minus of 10. So all values argued that than -10. So it will be just torn X Greater than -10. This will be some mission of all values that is zero, the one B two we three and before all values are satisfied this condition, so we had the total to well piers.

Here on this problem, we've been given a table which we're told is supposed to represent the probability mass function. Now we would like to verify that it is in fact a probability mass function. And then we want to find the required probabilities. Now in order to verify that it is a probability mass function, we need to have that each of the probabilities Is greater than or equal to zero and we need the some of the probabilities To equal one. These are two conditions for this to be a probability mass function. Now, the first one, you can clearly see that all of our epidemics values are greater than zero. And so that one holds. And then for the second one we want the probability of 1.25 Plus. The probability of 1.5 Plus, the probability of 1.75 Plus the probability of two Plus the probability of 2.25, We need that to equal one. So I point to, Well 0.4 Plus 0.1 Plus 0.2. 0.1. We need that to equal one. And it does. And so this is a probability mass function. Now to find a required probabilities on a. We want the probability That X is greater than or equal to two. Well this is equal to the probability that access to Plus the probability that X is 2.25. Because those were the only two values where the X value is two or larger Probability of 2 0.2. Our ability to 250.1. So this gives us 0.3 On B. We want the probability acts is less than 1.65. And so the only value is that we have there are less than 1.65 Are 1.5 And 1.25. So we're going to add those together. The probability of 1.5, I believe, 1.25, And so this feels it's a probability No 0.6 And see if we want the probability that X is 1.5. There's only one value on the table where X is 1.5 And we're told that probability is 0.4. And so that when we just needed to look at the chart in order And then on the we want to be less than 1.3 or we want X to be greater than 21 I'm Sorry, The only time X is less than 1.3 is at 1.25. The only time X is greater than 2.1 is at 2.25 Fact is 1.25. That's a probability of 0.2, And then the next is 2.25. That's a probability of 0.1, and it is together 0.3.

First problem. We've been given the probability mass function as of acts is equal 2, 3/4 times 1/4 raised to the X power or X 012 and then so on. So we have accountably infinite number of different values for first. We want to verify that this is a probability mass murder. Mhm. The first criteria is that each value of FX must be greater than or equal to zero and that is clearly true. Next we need to find the sum of all the F of X. Y. So putting in zero would give us three force Again in one would give us 3 16th three 64th and then so on. And you'll notice that this is an infinite geometric series whose first term is 3/4 and his common ratio is one for And so the formula for the sum of an infinite geometric series. His first term over one minus a common ratio. So this gives us 3/4 over 3/4 which is one. And so we do have that problem, that property. And probability mass function is verified. And so it is a probability mass function. Now let's find some probabilities on any. We want to find the probability that X is equal to two. Well, this is just equal to F two. So all we need to do is put into forex. And so this is 3/4 times 1/4 to the second, which gives us 3/64. And so the probability that access to There's 3/64 on B. We want to find the probability That X is less than or equal to two. Well this is equal to f of zero, Plus half of one Plus half of two. And so I plug in zero. We plug in one. Mhm. And we put it into uh oh me This is because it's 3/4 was 3 16, It was 3 64th. Yeah. Yeah. Which is 63/64. So that's the probability actually less than the flu. Okay. one C. We want the probability X is greater than two. And so using our complement rule, This is equal to 1-. The probability X is less than or equal to two. So this is 1 -63/64 Which is 1/64. And lastly on the we want the probability X is greater than or equal to one. This is 1- the probability that X is less than one. So this is one minus the probability x zero. Just 1 -3/4 Times 1/4 to the zero. Which gives us our probability of 1 4th

Given to a difference in this problem is three by four. one x 4 to the Power X. Mm. So yeah, this is a GP from to check whether it is a plurality funds and or not we have the first See submission acts starting for initial which is zero up to infinity. This plurality function should be the one in this case. Let's check. Yeah submission. That's a purchase. 0 to infinity three by four. When by 4 to the Power X will become by the formula GP form. It will be yeah. R to the power and -1 upon AR -1. This is Gps submission. Right? So Replacing here here he will use three x 4 Our is one x 4 and it's infinity And again AR -1 which is one x 4 -1. So this becomes three by four. Zero minus one Upon minus of three x 4. Which is against him -3 x four upon -3 x four. Your final values who want So we can we have to hence the well defense them. No, the other parties asking about the probabilities party, party X equal to so pearly X equal to do will be just put the value to in the function three before one x 4 XX values too. So it is just three x 64. 20 X. Less than it costs to. She had to be included. Zero what? Mhm 01 and two. Sorry, help all three will be included because equal to sign is also present. So this all three values three x 4. When they fall to the power-0 Last three x 4. one x 4. to the power one plus three x 4. one x 4. to the power to This will be three x 4 plus three by 16 plus three. Right, 64. So overall it will become 48 plus well last week by 64. So it will be finally 63 x 64. Well the x greater than two will be then just 1 -20 x less than equal to So 1- Just about Calculated values 63 or 64. This will be won by 64 and the party. Mhm Girl X Better than equal to one Is because to 1 -20 X equal to zero. This can be written as 1 -1 three x 4 or we can say final answer ability one x 4. So this is the complete solution.


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