Find Cov(Z,W) with E(X^2)if X is N(0,1)

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In summary: The chi-squared distribution is a special case of the gamma distribution and is one of the most widely used probability distributions in inferential statistics, notably in the analysis of variance and in hypothesis testing.In summary, the distribution of Y, where Y=X^2 and X is a standard normal random variable, is the chi-squared distribution with 1 degree of freedom. This distribution is commonly used in inferential statistics and hypothesis testing.
  • #1
WMDhamnekar
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Let X and Y be two independent [Tex]\mathcal{N}(0,1)[/Tex] random variables and

[Tex] Z=1+X+XY^2[/Tex]

[Tex]W=1+X[/Tex]
I want to find Cov(Z,W).

Solution:-

[Tex]Cov(Z,W)=Cov(1+X+XY^2,1+X)[/Tex]

[Tex]Cov(Z,W)=Cov(X+XY^2,X)[/Tex]

[Tex]Cov(Z,W)=Cov(X,X)+Cov(XY^2,X)[/Tex]

[Tex]Cov(Z,W)=Var(X)+E(X^2Y^2)-E(XY^2)E(X)[/Tex]

[Tex]Cov(Z,W)=1+E(X^2)E(Y^2)-E(X)^2E(Y^2)[/Tex]

[Tex]Cov(Z,W)=1+1-0=2[/Tex]

Now E(X)=0, So [Tex]E(X)^2E(Y^2)=0[/Tex], But i don't follow how [Tex]E(X^2)E(Y^2)=1?[/Tex] Would any member explain that? My another question is what is [Tex]Var(X^2)?[/Tex]
 
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  • #2
Re: E(x^2)and VAR(x^2)if X is N(0,1).

Dhamnekar Winod said:
My another question is what is [Tex]Var(X^2)?[/Tex]

Hi Dhamnekar,

Let's start with this one.
It's the variance. It is the mean of the squared deviations from the average.
And the average of $X$ is the same thing as the expected value $E(X)$ or just $EX$.
In formula form:
$$\operatorname{Var}(X) = E\left((X - EX)^2\right)$$
If we write it out, we can find that it can be rewritten as:
$$\operatorname{Var}(X) = E(X^2) - (EX)^2$$

Dhamnekar Winod said:
Let X and Y be two independent [Tex]\mathcal{N}(0,1)[/Tex] random variables and

(snip)

Now E(X)=0, So [Tex]E(X)^2E(Y^2)=0[/Tex], But i don't follow how [Tex]E(X^2)E(Y^2)=1?[/Tex] Would any member explain that?

Now let's get back to your first question.

The fact that $X \sim \mathcal{N}(0,1)$ means that $\operatorname{Var}(X)=1$.
Combine it with $EX=0$ and fill it in:
$$\operatorname{Var}(X) = E(X^2) - (EX)^2 \implies 1=E(X^2)-0 \implies E(X^2)=1$$
 
  • #3
Re: E(x^2)and VAR(x^2)if X is N(0,1).

Klaas van Aarsen said:
Hi Dhamnekar,

Let's start with this one.
It's the variance. It is the mean of the squared deviations from the average.
And the average of $X$ is the same thing as the expected value $E(X)$ or just $EX$.
In formula form:
$$\operatorname{Var}(X) = E\left((X - EX)^2\right)$$
If we write it out, we can find that it can be rewritten as:
$$\operatorname{Var}(X) = E(X^2) - (EX)^2$$
Now let's get back to your first question.

The fact that $X \sim \mathcal{N}(0,1)$ means that $\operatorname{Var}(X)=1$.
Combine it with $EX=0$ and fill it in:
$$\operatorname{Var}(X) = E(X^2) - (EX)^2 \implies 1=E(X^2)-0 \implies E(X^2)=1$$

Hello,
If $X$ be $\mathcal{N}(0,1)$ random variable, and $Y=X^2$ is the function of $X$, what is the distribution of $Y$?Is its distribution Normal?
 
  • #4
Re: E(x^2)and VAR(x^2)if X is N(0,1).

Dhamnekar Winod said:
Hello,
If $X$ be $\mathcal{N}(0,1)$ random variable, and $Y=X^2$ is the function of $X$, what is the distribution of $Y$?

Is its distribution Normal?

No...

In probability theory and statistics, the chi-squared distribution (also chi-square or $χ^2$-distribution) with $k$ degrees of freedom is the distribution of a sum of the squares of $k$ independent standard normal random variables.
 

FAQ: Find Cov(Z,W) with E(X^2)if X is N(0,1)

1. What is the formula for finding Cov(Z,W)?

The formula for finding Cov(Z,W) is Cov(Z,W) = E(ZW) - E(Z)E(W), where E(ZW) is the expected value of the product of Z and W, and E(Z) and E(W) are the expected values of Z and W, respectively.

2. How do you calculate E(X^2) for a normal distribution?

For a normal distribution with mean μ and standard deviation σ, the formula for calculating E(X^2) is E(X^2) = μ^2 + σ^2. In this case, since X is N(0,1), E(X^2) = 0^2 + 1 = 1.

3. Can you explain the concept of covariance in simpler terms?

Covariance measures the relationship between two variables. A positive covariance indicates that the variables tend to move in the same direction, while a negative covariance indicates that they tend to move in opposite directions. A covariance of 0 means that there is no relationship between the variables.

4. What does it mean to find Cov(Z,W) with E(X^2) if X is N(0,1)?

In this context, finding Cov(Z,W) with E(X^2) means that we are calculating the covariance between two variables, Z and W, using the expected value of X^2, where X is a normal distribution with mean 0 and standard deviation 1.

5. How is the value of Cov(Z,W) affected by the value of E(X^2)?

The value of E(X^2) does not directly affect the value of Cov(Z,W). However, since E(X^2) is used in the formula for calculating Cov(Z,W), a change in its value can result in a change in the value of Cov(Z,W).

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