Okay, So we're giving a probability distribution which I have written at the top here, and we're trying to find the mean value and the standard deviation for this probability distribution. So in order to find the okay, in order to find the the mean of our probability distribution, we have to find the expected value of X. It is equal to the expected value of X. And the formula to find that is just the summation From X is equal to zero to whatever our last X value is. In this case it's for and we multiply our X value times our probability at that X value. And so now we're just gonna plug in some of our X values and probabilities into this equation and find our view of X. So this is equal to zero times 0.5 plus one, Times .15 Plus two Times .25 for us three Times .25 plus four times 0.3. And if we add this all together we'll see that this is equal to 2.6 now to find the standard deviation? Well, the formula for that is the square root of our expected value of x squared minus the expected value of x squared. And that can get a little tricky. But you just have to remember the first one is, the X squared is actually within this expected value, and the second one is, we're actually just taking are expected value of X, whatever that number is, and then squaring it. So find the expected value of X squared, We have to find the summation from X is equal 0-4 of X squared times are probability at that X value. So this is gonna cool zero squared times five plus one squared, which is just one times point oh 0.15 Plus two squared which is four Times .25, Yeah plus plus three squared which is nine times .25 Plus four squared which is 16 times 0.3. You know, if we have this all together we're going to get a value of 8.2. Now we have to do is plug this back into our equation and solve for our standard deviation. So we have the standard deviation is equal to 8.2 -2.6 squared. All square rooted. So if you plug this into our calculators, We'll see that this is equal to 1.2. Mhm. And now for part B we're trying to find again is a you are mean and a standard deviation Except this time we're trying to find the mean value and standard deviation of 150 students buying tickets with the probability that we have above. So in order to find the mean value, we can use this formula, which is the main value is our main value of our probability distribution Times our end value, which is in this case 150. So we get 2.6 Times 150, Which is equal to 390. Now to find the standard deviation, What we can do is again use a formula using the standard deviation we found in part A which is our standard deviation, found in part a Multiplied by the square root of our end value, which again in this case is 150, So we have 1.2 times the square root Of 150. And if we plugged that into a calculator We get 150, Sorry, the square root of 150 Time to 1.2, which is equal to around 14.7. Yeah. And now for part C, What we're trying to find is the probability that given that we have a maximum amount of 500 seats available, that 150 people buying tickets will all be accommodated for. So for every single one of the students to be accommodated for what we need is the Amount of tickets bought to be less than or equal to 500. So we're trying to find the probability that our tea Which is the amount of tickets bought by 150 students is less than or equal to 500. And so I'm going to do this is by using a Z. Score and transforming our probabilities so that we can use the standard normal distribution table to find our probabilities. So how I do this is we have Z is equal to t minus our U. of T value, all divided by our standard deviation that we found in part B. So if we do this, we have t, which is 500 minus UFT is 390 Divided by our standard deviation, which is 14.7. So if we plug this into a calculator, we get 110 divided by 14.7, which is equal to 7.4 eight. So now we're trying to find the probability that he is, sorry, that Z Is less than or equal to 7.48. And if you look at this already, we can see that some .48 is a massive value for a standard normal um distribution table. So This is going to be around .9999. Given that our standard deviation is 14.7 and we want to have a maximum number students of 500 and are mean number of students is 390, it kind of makes sense for the probability to be insanely high