Expected Value, Expected Variance,covariance

In summary: Similarly, the notation for variance is$$\text{var}(X) = E\{X^2\} - E\{X\}^2 = \frac {\sum_{X = -1}^1 X^2 p(X) }{\sum_{X = -1}^1 p(X)} - \left(\frac {\sum_{X = -1}^1 X p(X) }{\sum_{X = -1}^1 p(X)} \right)^2$$And your covariance is $$\text{cov}(X, Y) = E\{(X - E\{X\})(Y - E\{Y\})\} = \frac {\sum_{
  • #1
prodicalboxer
1
0
Can someone help me with this problem?

The joint probability mass function of X and Y, p(i,j)=P{X=i,Y=j}, is given as follows
p(-1,-2)=1/9, p(-1,-1)=1/18, p(-1,0)=1/12, p(-1,1)=0,
p(0,-2)=1/12, p(0,-1)=1/9, p(0,0)=0, p(0,1)=1/8,
p(1,-2)=0, p(1,-1)=1/8, p(1,0)=1/4, p(1,1)=1/18,

a) Compute the E[X], Var(X), and Cov(X,Y)
b) Calculate P{X,Y=k} for k=-2,-1,0,1,2
c) Evaluate E[Y|X=k] for k=-1,0,1

here is what I attempted to do:
E[X]=E[X1] + E[X2]+....E[Xn]=np
Var(X)= E[X^2]-(E[X])^2
Cov=(X,Y)=E[(X-E[X])(Y-E[Y])]
=E[XY-YE[X]-XE[Y]+E[X]E[Y]]
=E[XY]-E[Y]E[X]-E[X]E[Y]+E[X]E[Y]
=E[XY]-E[X]E[Y]

E[X]=E[X|Y=-2]= 1/9(-1)+1/12(0)+0(1)=-1/9
E[X|Y=-1]=1/18(-1)+1/9(0)+1/8(1)=5/72
E[X|Y=0]=1/12(-1)+0(0)+1/4(1)=1/6
E[X|Y=1]=0(-1) + 1/8(0) +1/18(1)=1/18
E[X]=-1/9+5/72+1/6+1/18=13/72

now the variance Var(X)
(-1/9)^2+(5/72)^2+(1/6)^2+(1/18)^2=1/81+25/5184+1/36+1/324
=83/1728
Var=83/1728-169/5184=5/324

E[Y]= [Y|X=-1]=1/9(-2)+1/18(-1)+1/12(0)+0(1)=-5/18
[Y|X=0]=1/12(-2)+1/9(-1)+0(0)+1/8(1)=-11/72
[Y|X=1]=0(-2)+1/8(-1)+1/4(0)+1/18(1)=-5/72
E[Y]=-5/18 -11/72-5/72=-41/72

Cov=(83/1728x5/324)-(13/72x-41/72)=37/5000

Could you check part A ...I really need help with part b and c
 
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  • #2
Your numerical answer is correct for for ##E\{X\}##, but if I were grading your homework, I might mark off a point for your notation.
$$E\{X|Y=-2\} = \frac {\sum_{X = -1}^1 X p(X, -2)}{\sum_{X = -1}^1 p(X, -2)}$$
so you shouldn't use the notation ##E\{X|Y=-2\}## to mean something else. What you can do is write
$$E\{X\} = {\sum_{X = -1}^1 \sum_{Y = -2}^2X p(X, Y)}$$
and that is what you ended up doing when you calculated the numerical result.
 

1. What is the expected value and how is it calculated?

The expected value is a measure of the central tendency of a random variable. It is calculated by multiplying each possible outcome by its corresponding probability, and then summing all the values.

2. How is expected value useful in decision making?

Expected value can be used in decision making to assess potential risks and rewards. By calculating the expected value of different outcomes, decision makers can make informed choices based on the potential outcomes.

3. What is expected variance and how is it related to expected value?

Expected variance is a measure of the spread or variability of a random variable. It is calculated by taking the sum of squared differences between each possible outcome and the expected value, multiplied by their corresponding probabilities. Expected variance is related to expected value because it provides additional information about the distribution of possible outcomes.

4. What is covariance and how is it related to expected value and expected variance?

Covariance is a measure of the relationship between two variables. It indicates how much two variables change together. It is related to expected value and expected variance because it takes into account the expected values and expected variances of both variables in its calculation.

5. How is covariance useful in data analysis?

Covariance is useful in data analysis because it can help identify patterns and relationships between variables. It is commonly used in statistical models and can provide valuable insights into the data being analyzed.

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