Let X and Y have the joint pmf p(x, y)=e^-2 /[x !(y-x) !], y=0,1,2, …, x=0,1, …, y, zero elsewhe

Let X and Y have the joint pmf p(x, y)=e^-2 /[x !(y-x) !],...

Key Concepts

Joint Probability Mass Function (pmf)

The joint pmf describes the probability distribution of two discrete random variables defined on a common sample space. It assigns probabilities to pairs of outcomes, and is essential for understanding how these variables interact and influence one another. In general, the joint pmf is the starting point for computing marginal distributions, conditional probabilities, and various moments of the random variables.

Moment Generating Function (MGF)

The MGF is a function that encapsulates all of the moments (expected values of powers) of a random variable or a vector of random variables. For joint distributions, the joint MGF is defined as the expected value of the exponential of a linear combination of the variables. This function is a powerful tool for deriving means, variances, covariances, and identifying the distribution of sums of independent random variables.

Marginal Distributions and Moments

Marginal distributions are obtained by summing (or integrating, in the continuous case) the joint pmf over one of the variables, giving the distribution of the remaining variable alone. The moments, such as the mean and variance, of a random variable are computed using its marginal distribution. These moments are critical for quantifying the central tendency and the dispersion of the distribution.

Covariance and Correlation Coefficient

Covariance measures the degree to which two random variables change together, while the correlation coefficient standardizes this measure to a value between -1 and 1, providing a scale-free indication of linear dependence. These measures are essential for understanding the relationship between random variables and are computed using their expectations and variances.

Conditional Expectation

Conditional expectation is the expected value of one random variable given the value of another. It provides insight into how the distribution of one variable behaves under the condition that another variable takes on a specific value. This is a key concept in probability theory and statistics, especially in the analysis of dependent random variables.

Binomial Theorem and Combinatorial Identities

The binomial theorem is a fundamental principle in algebra that describes the expansion of powers of a sum, and it frequently appears in probability theory in the context of combinations or binomial events. Combinatorial identities, such as those involving factorials and combination coefficients, are used to simplify expressions in probability, especially when working with discrete distributions that have combinatorial structures.

Transcript

00:01 Hi, here it has been given that x and y has joined p mf that is p xy, p xy.

00:10 This is equals to probability capital x equals to small x intersection, capital y equals to small y and this is equal to e to 1 minus 2 whole divided by x factorial y minus x factorial such that x equals to 0 1 2 up to y and y equals to 0 1 2 and so on and 0 otherwise first we have to find we have to find the moment generating function of x y that is denoted by m t1 t2 second you have to find the mean and variance of x and y the expectation x variance x expectation x variance x expectation y variance y okay now third also second question another part is there that is we'll find the coalition coefficient between x y and lastly we have to find the expected value of x given y okay now let's start with the problem so first is moment generating function of x y that is m t1 comma t2 equals to expectation of e to the power t1x plus t2y right this is equals to summation y runs from 0 to infinity summation x runs from 0 to y e to the power t 1x plus t 2 small y is e to the power of minus 2 whole divided by x factorial into y minus x factorial because expectation means value into probability on this e to the 4 minus 2 comes out summation y runs from 0 to infinity, x transforms 0 to y.

02:07 Now, e to the 4, t1x plus t2y, whole divided by we are writing y factorial in the denominator and y factorial in numerator.

02:22 So that it both cancels.

02:27 Now, this is y combination c.

02:30 Now it can return at y a to equal minus two now y part we write with this summation okay it will be e to the power t2i whole divided by y factorial now summation x runs from 0 to y a to the 4 t1 x and this is y combination c see this is a binomial expansion right because we know that a plus b whole to the power n equals to summation r runs from 0 to n, ncr, b to the power r, a to the power n minus r.

03:12 From here, n we have y, right? r we have taken as x and b is e to the power t1 and a is 1.

03:25 So it will be e to the power minus 2 summation, y runs from 0 to infinity, a to the power 2, 2y all divided by y factorial into 1 plus e to the 4 t 1 whole to the power y right so this is equals to a to the power minus 2 summation y runs from 0 to infinity e to the power t 2 into 1 plus e to the power t 1 this whole to the core y 4 divided by y factor now this is a summation of exponential series because we know that e to the power x equals to summation r runs from 0 to infinity x to the power r all divided by r factor here right so this will be equals to e to the power minus 2 e to the power e to the power t 2 this will be in bracket 1 plus e to the power t 1 right so finally we have the moment generating function m m p 1 comma t 2 2 equals to e to the power minus 2 into e to the power e to the power t 2 into 1 plus e to the power t 1 2 bracket close when t 1 comma t 2 not equals to 0 and 1 when t1 t 1 equals to 0 equals to t 2 this is our moment genetic function now going forward to the second problem we need to find the million variance of x and 1 and for that we need to first we have to find the marginal probability mass function of x and y.

05:15 Okay now the marginal probability mass function of y that is probability capital y equals to small y we write it small p y this will be the range of x now x runs from 0 to y so it will be x equals to 0 to y p xy the joint probability mass function p xy this will be summation x runs from 0 to y e to the power minus 2 whole divided by x factorial into y minus x factor okay now e to the power minus 2 comes out and we are writing in the denominator y factorial and the numerator also y factorial x equals to 0 to y this will be y factorial whole divided by x factorial into y minus x factorial so this is y combination x now this is also a binomial expansion where a and b is 1 to the power x and 1 to the power y minus x because this one to the power x and one to the power minus total is 1 so we can write it like that so it will be to the power minus 2 whole divided by y factorial into 2 to the power 1 plus 1 so 2 to the power y so this is so p y equals to e to the power minus 2 2 to equal y whole divided by y factorial where y equals to 0 1 2 and so on so clearly from here we can write that y follows poissure distribution of parameter 2 okay so from here it is evident that expectation of y equals to 2 and variance of y is also equals to 2 because mean and variance are exactly same and that is exactly same as the parameter of the poised distribution now we'll find the million variance of x.

07:06 So for that, we need to first we have to find the marginal probability mass function of x, that is px, it will be range of y.

07:15 Right...

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