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    (Original post by ServantOfMorgoth)
    I disagree I wished I had more experience with statistics at a level to get the basics down. I've had to start from square 1 and learn it in the equivalent of say S1 - S4 only 3 or so months. It's really hard.
    I have a lot of sympathy with what you say and I see not a few students in the same position as you. Let me generalize away from the struggle you are having, and pontificate on the generality. Please don't think I am picking on you!

    To me there seem to be a couple of underlying problems that students arriving at university have when they get thrown in at the deep end with probability and statistics - especially those of you whose are not specializing in mathematics: (i) what they have been taught appears as a rag-bag of only vaguely connected ideas; there's very little solid understanding of what is really going on here; (ii) there's not enough facility with the basics - algebra and calculus, in particular.

    Now, compounding this is how "service course" probability and statistics is taught at university. This makes me shudder.

    Hence my emphasis on "the basics". If I did have my way, I would probably replace a lot of school and first year service P&S with "1001 things you can do with a random number generator" - focusing on the underlying ideas using modern computational statistics as the framework. This latter seems to be something completely invisible at school level, and yet is fundamental in real statistics.

    FM students would also get "1001 things you can do with generating functions and stochastic matrices", but that's another matter.
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    (Original post by Gregorius)
    I have a lot of sympathy with what you say and I see not a few students in the same position as you. Let me generalize away from the struggle you are having, and pontificate on the generality. Please don't think I am picking on you!

    To me there seem to be a couple of underlying problems that students arriving at university have when they get thrown in at the deep end with probability and statistics - especially those of you whose are not specializing in mathematics: (i) what they have been taught appears as a rag-bag of only vaguely connected ideas; there's very little solid understanding of what is really going on here; (ii) there's not enough facility with the basics - algebra and calculus, in particular.

    Now, compounding this is how "service course" probability and statistics is taught at university. This makes me shudder.

    Hence my emphasis on "the basics". If I did have my way, I would probably replace a lot of school and first year service P&S with "1001 things you can do with a random number generator" - focusing on the underlying ideas using modern computational statistics as the framework. This latter seems to be something completely invisible at school level, and yet is fundamental in real statistics.

    FM students would also get "1001 things you can do with generating functions and stochastic matrices", but that's another matter.
    For mechanical engineering all the maths I did has been very useful. I think if I hadn't done all mechanics modules I'd have likely struggled because I've seen some of my mates struggle with the mechanics because they didn't do as much as the rest of us.
    Maybe you're looking at it from a pure maths point of view because I felt like if I had done S modules at a level I'd have been fine now. Like some of my mates that did stats modules are doing so much better than me with it now.

    I know this may make me look stupid (as if this thread hasn't already enough), too much background theory is usually very confusing. Like when we did vector calculus some of the definitions and proofs were so confusing I never understood them nor did most people (I think) but doing the questions turned out to be easier (not too easy though).

    For stats my lectures just teach the theories of statistcs with maybe 1 example per topic and it's very confusing trying to apply those highly convoluted mathematical prose into a format for solving questions.
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    (Original post by ServantOfMorgoth)
    I know this may make me look stupid (as if this thread hasn't already enough), too much background theory is usually very confusing.
    I promise you, it doesn't make you look stupid! There are many students in similar positions to you. I think one problem is that this "background theory" is introduced in such an unmotivated way - and it needn't be like this. There are a handful of foundational principles in P&S at this level - and motivating them is not difficult, especially if modern simulation methods are used.

    Anyway, keep firing off the questions!
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    (Original post by Gregorius)
    I promise you, it doesn't make you look stupid! There are many students in similar positions to you. I think one problem is that this "background theory" is introduced in such an unmotivated way - and it needn't be like this. There are a handful of foundational principles in P&S at this level - and motivating them is not difficult, especially if modern simulation methods are used.

    Anyway, keep firing off the questions!
    From Monday I'll be working on stats solely but for now I have a lot of other things going on and I was just trying to squeeze it in.

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    (Original post by ServantOfMorgoth)
    For mechanical engineering all the maths I did has been very useful. I think if I hadn't done all mechanics modules I'd have likely struggled because I've seen some of my mates struggle with the mechanics because they didn't do as much as the rest of us.
    Did you do M1-5 for A-level? Is it worth self-teaching concepts from M4 or M5 during my summer?
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    (Original post by aymanzayedmannan)
    Did you do M1-5 for A-level? Is it worth self-teaching concepts from M4 or M5 during my summer?
    No not really. It'll be really hard to self teach and engineering is tough, enjoy the free time while you have it.
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    (Original post by Gregorius)
    I promise you, it doesn't make you look stupid! There are many students in similar positions to you. I think one problem is that this "background theory" is introduced in such an unmotivated way - and it needn't be like this. There are a handful of foundational principles in P&S at this level - and motivating them is not difficult, especially if modern simulation methods are used.

    Anyway, keep firing off the questions!

    Marginals in statistics. I don't understand them and my notes have nothing on it so I don't exactly know what I'm looking for or what's above the level I need to know. So I have a few questions.

    What is the definition of a marginal pdf?

    Based on how the function looks I know that I have to do double integration. I have a question with a solution but I'm not sure what they did.

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    (Original post by ServantOfMorgoth)
    Marginals in statistics. I don't understand them and my notes have nothing on it so I don't exactly know what I'm looking for or what's above the level I need to know. So I have a few questions.

    What is the definition of a marginal pdf?
    There are three concepts you have to get your head around: the joint probability function, the conditional probability function and the marginal probability function.

    If you have two random variables (X and Y), then the joint probability function simply describes the probability of the co-occurence of values. If we stick to discrete RVs for the moment with X taking on possible values a1, ..., an and Y taking on possible values b1,...,bm, the the joint probability function gives you the probability that, at the same time, X=ai and Y=bj.

    The conditional probability function gives you the probability that X=ai given that we already klnow that Y=bj. If you think geometrically, then this distribution will be a slice through the joint probability distribution at Y=bj, scaled so that the sum over possible X values is unity.

    The marginal probability distribution gives you the probability distribution of X irrespective of what's happening with Y. So to get the marginal probability distribution for any particular value of X (= aj, for instance) you sum up the values of the joint probability distribution over all X, Y where X=aj.

    When you turn to continuous probability distributions, sums become integrals and the definition of marginal probability becomes

    \displaystyle f_X(x) = \int f_{X, Y} (x,y) dy

    where the integral is taken over all possible values of Y.

    Based on how the function looks I know that I have to do double integration. I have a question with a solution but I'm not sure what they did.
    So for your example, the marginal probability function of X is given by

    \displaystyle f_X(x) = \int x e^{-x(1+y)} dy = e^{-x}

    So if you are just working with values of X irrespective of the values of Y, use the marginal probability distribution. Can you take it from there? (Sorry, can't read your attached answer).
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    (Original post by Gregorius)
    There are three concepts you have to get your head around: the joint probability function, the conditional probability function and the marginal probability function.

    If you have two random variables (X and Y), then the joint probability function simply describes the probability of the co-occurence of values. If we stick to discrete RVs for the moment with X taking on possible values a1, ..., an and Y taking on possible values b1,...,bm, the the joint probability function gives you the probability that, at the same time, X=ai and Y=bj.

    The conditional probability function gives you the probability that X=ai given that we already klnow that Y=bj. If you think geometrically, then this distribution will be a slice through the joint probability distribution at Y=bj, scaled so that the sum over possible X values is unity.

    The marginal probability distribution gives you the probability distribution of X irrespective of what's happening with Y. So to get the marginal probability distribution for any particular value of X (= aj, for instance) you sum up the values of the joint probability distribution over all X, Y where X=aj.

    When you turn to continuous probability distributions, sums become integrals and the definition of marginal probability becomes

    \displaystyle f_X(x) = \int f_{X, Y} (x,y) dy

    where the integral is taken over all possible values of Y.



    So for your example, the marginal probability function of X is given by

    \displaystyle f_X(x) = \int x e^{-x(1+y)} dy = e^{-x}

    So if you are just working with values of X irrespective of the values of Y, use the marginal probability distribution. Can you take it from there? (Sorry, can't read your attached answer).
    Yeah, thanks a lot, I understood how to do the first part, but if you only used partial integration in the first part why do you have to use a double integral to find  P(X \geq 3) ?
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    (Original post by ServantOfMorgoth)
    Yeah, thanks a lot, I understood how to do the first part, but if you only used partial integration in the first part why do you have to use a double integral to find  P(X \geq 3) ?
    Not sure what you mean by a "partial integral". But to get the marginal pdf of X for part (i), you integrate out the Y to get the distribution f_X(x) = e^{-x}. You then work directly with this function to answer part (ii). That is, you evaluate

    \displaystyle \mathbb{P}(X \ge 3) =  \int_{3}^{\infty} e^{-x} dx
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    (Original post by Gregorius)
    Not sure what you mean by a "partial integral". But to get the marginal pdf of X for part (i), you integrate out the Y to get the distribution f_X(x) = e^{-x}. You then work directly with this function to answer part (ii). That is, you evaluate

    \displaystyle \mathbb{P}(X \ge 3) =  \int_{3}^{\infty} e^{-x} dx
    By partial integration I meant that x was treated as a constant since the integral was wrt to y.

    And oh, in the solution, they started again with the definition of the marginal pdf but with a double integral instead with limits 0 to infinity for the y integral and 3 to infinity for the x integral. But looking at it now, I see that it would reduce back down to this what you have here. Thanks again.
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    (Original post by Gregorius)
    Not sure what you mean by a "partial integral". But to get the marginal pdf of X for part (i), you integrate out the Y to get the distribution f_X(x) = e^{-x}. You then work directly with this function to answer part (ii). That is, you evaluate

    \displaystyle \mathbb{P}(X \ge 3) =  \int_{3}^{\infty} e^{-x} dx

    I feel like I'm doing something wrong here?

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    EDIT

    I just saw my error.


    EDIT again

    I have no idea how to do  P( Y< 2X) and  P(X^2 + Y^2 < 1) or part C either. Also how would you know if X and Y are independent?

    Gregorius Zacken
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    (Original post by ServantOfMorgoth)
    I feel like I'm doing something wrong here?

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    EDIT

    I just saw my error.


    EDIT again

    I have no idea how to do  P( Y< 2X) and  P(X^2 + Y^2 < 1) or part C either. Also how would you know if X and Y are independent?

    Gregorius Zacken
    Indeterminate do you have any idea how to do these?
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    (Original post by ServantOfMorgoth)

    I just saw my error.
    OK, so I take it that you've spotted the error in your initial double integral over the unit square; so you should have arrived at the value of 3 for c.

    EDIT again

    I have no idea how to do  P( Y< 2X) and  P(X^2 + Y^2 < 1) or part C either. Also how would you know if X and Y are independent?
    You are given the joint probability density. So, to work out P(Y < 2x), you notice that this is the same as P(2X - Y > 0). Therefore you need to work out the integral of the joint probability density over the part of the unit square in the X-Y plane where 2X - Y > 0. It's an exercise in getting the limits of a double integral correct!

    The same is true for the next part. You need to find the points {(X,Y); X^2 + Y^2 < 1} in the unit square and integrate the joint probability density over this region. Again, an exercise in getting the limits of a double integral right.

    Part (c), you have the correct approach in your jottings. Integrate (x+y) times the joint density function over the unit square.

    In part (d) you're asked to calculate the marginal density functions of X and Y respectively, so to finish off the question about independence, you recall that X and Y are independent if and only if their joint probability density function is equal to the product of the marginal density functions for X and Y.
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    (Original post by Gregorius)
    OK, so I take it that you've spotted the error in your initial double integral over the unit square; so you should have arrived at the value of 3 for c.



    You are given the joint probability density. So, to work out P(Y < 2x), you notice that this is the same as P(2X - Y > 0). Therefore you need to work out the integral of the joint probability density over the part of the unit square in the X-Y plane where 2X - Y > 0. It's an exercise in getting the limits of a double integral correct!

    The same is true for the next part. You need to find the points {(X,Y); X^2 + Y^2 < 1} in the unit square and integrate the joint probability density over this region. Again, an exercise in getting the limits of a double integral right.

    Part (c), you have the correct approach in your jottings. Integrate (x+y) times the joint density function over the unit square.

    In part (d) you're asked to calculate the marginal density functions of X and Y respectively, so to finish off the question about independence, you recall that X and Y are independent if and only if their joint probability density function is equal to the product of the marginal density functions for X and Y.
    So the statistics exam went really good. Thanks a lot for all your help.😊.
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    (Original post by ServantOfMorgoth)
    So the statistics exam went really good. Thanks a lot for all your help.😊.
    Hey, that's really good. Well done!
 
 
 
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