Sum/differences of two normal variables

In summary, the mean of Bt+s + Bt+s - Bs - Bs is 0 and the variance is 2t+4s. This is because the sum of independent normals is normal with mean = sum of means and variance = sum of variances. Additionally, the property (ii) states that cX ~ N(cμ, c2σ2) regardless of the sign of c. Therefore, the negative sign in front of Bs can be disregarded when calculating the variance.
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Homework Statement



I have Bt+s ~ N (0, t+s) and Bs ~ N(0,s).
I want to find the mean and variance of Bt+s + Bt+s - Bs - Bs.

Homework Equations



(i) I know that the sum of independent normals is again normal with mean = sum of means and variance = sum of variances.

(ii) I also know that cX ~ N(cμ, c2σ2)

The Attempt at a Solution



Obviously the mean is easy to calculate since it is 0 all round, but I'm having problems with the variance.

If I wanted to find distribution of:
Bt+s + Bt+s it would be ~N(0,t+s+t+s)
No issues here

But now since I have a negative infront of the Bs, how will that come into effect? Notice how property (i) above is regarding the sum of independent normals, what about the difference between two independent normals?

Do I change -Bs (by property (ii) above) so that it has a distribution ~ N ((-1)0, (-1)2(s))

Does this change the negative out the front of Bs to a positive?

i.e, -Bs ~ N (0, s) and then sum the variances so that:

Bt+s + Bt+s - Bs - Bs ~ N (0, (t+s)+(t+s)+(s)+(s))

i.e. Bt+s + Bt+s - Bs - Bs ~ N (0, 2t+4s)?

hope it all makes sense
 
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  • #2
If [itex] X [/itex] is [itex] n(\mu, \sigma^2) [/itex]

then [itex] aX[/itex] is [itex]n(a\mu, a^2 \sigma^2)[/itex]

If [itex] X, Y [/itex] are independent, then

[tex]
\sigma^2_{aX+bY} = a^2 \sigma^2_X + b^2\sigma^2_Y
[/tex]

and both of these hold regardless of the sign of [itex] a [/itex] and [itex] b [/itex].
 

1. What is a normal distribution?

A normal distribution is a probability distribution that is often used to describe real-world phenomena, such as height or IQ scores. It is characterized by a bell-shaped curve and is symmetrical around the mean.

2. How do you calculate the sum of two normal variables?

To calculate the sum of two normal variables, you simply add their means and variances. This is known as the "sum rule" for normal distributions.

3. What is the difference between adding two normal variables and multiplying two normal variables?

Adding two normal variables results in a normal distribution with a larger mean and variance, while multiplying two normal variables results in a normal distribution with a larger mean and a much larger variance. In other words, addition increases the spread of the distribution, while multiplication increases both the spread and the center.

4. Can you combine normal variables with different means and variances?

Yes, you can combine normal variables with different means and variances. This is known as the "sum rule" for normal distributions and is commonly used in statistics and probability calculations.

5. How do normal variables affect each other when combined?

When normal variables are combined, they may interact in different ways depending on their means and variances. For example, if two normal variables have similar means and variances, their combination will result in a larger normal distribution with a similar shape. However, if two normal variables have very different means and variances, their combination may result in a skewed or non-normal distribution.

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