When the mean of a Gaussian distribution is 𝜇 μ and the variance is 𝜎 2 σ 2 , the expression for its probability density function 𝑓 𝜇 , 𝜎 2 f μ,σ 2 ​ is: 𝑓 𝜇 , 𝜎 2 ( 𝑡 ) = 1 2 𝜋 𝜎 2 𝑒 − ( 𝑡 − 𝜇 ) 2 2 𝜎 2 f μ,σ 2 ​ (t)= 2πσ 2 ​ 1 ​ e − 2σ 2 (t−μ) 2 ​ Specifically, the Fourier transform of the Gaussian is given by: 𝑓 ^ 0 , 𝜎 2 ( 𝜈 ) = 𝑒 − ( 2 𝜋 𝜈 ) 2 𝜎 2 / 2 f ^ ​ 0,σ 2 ​ (ν)=e −(2πν) 2 σ 2 /2 For the convolution, the following holds: 𝑓 0 , 0.7 ∗ 𝑓 0 , 0.3 = 𝑓 0 , 𝑐 2 f 0,7 ​ ∗f 0,3 ​ =f 0,c 2 ​ Calculate the corresponding standard deviation 𝑐 = 𝑐 2 c= c 2 ​ .

The problem asks to find the corresponding standard deviation \( c \) when two Gaussian distributions \( f_{0, 0.7} \) and \( f_{0, 0.3} \) are convolved, and the result is a Gaussian distribution \( f_{0, c^2} \).

In general, when convolving two Gaussian distributions with variances \( \sigma_1^2 \) and \( \sigma_2^2 \), the variance of the resulting distribution is the sum of the variances:

\[
\sigma_{\text{result}}^2 = \sigma_1^2 + \sigma_2^2
\]

Given the two variances in this case:

- \( \sigma_1^2 = 0.7 \)
- \( \sigma_2^2 = 0.3 \)

The resulting variance \( c^2 \) is:

\[
c^2 = 0.7 + 0.3 = 1.0
\]

Now, to find the standard deviation \( c \), we take the square root of the variance:

\[
c = \sqrt{c^2} = \sqrt{1.0} = 1.0
\]

Thus, the corresponding standard deviation is \( c = 1.0 \).

Would you like more details on the convolution of Gaussian distributions or any other aspect?

Here are five follow-up questions you might find useful:
1. What is the convolution of two Gaussian functions and why does it result in another Gaussian?
2. How does the Fourier transform relate to Gaussian functions?
3. Can you explain the properties of the Gaussian distribution that make it so widely used in statistics?
4. How does the variance of a Gaussian affect the shape of its probability density function?
5. What are some practical applications of convolving Gaussian functions?

**Tip:** When working with convolutions of Gaussians, always remember that variances add, which simplifies many problems in probability and signal processing.

When the mean of a Gaussian distribution is μ and the variance is σ^2, the expression for its probability density function f(μ,σ^2) is: f(μ,σ^2)(t) = (1/√(2πσ^2)) * e^(-(t−μ)^2 / (2σ^2)). Specifically, the Fourier transform of the Gaussian is given by: f̂(0,σ^2)(ν) = e^(-(2πν)^2σ^2 / 2). For the convolution, the following holds: f(0,0.7) * f(0,0.3) = f(0,c^2). Calculate the corresponding standard deviation c = √c^2.

Learn how to calculate the standard deviation when convolving two Gaussian distributions f(0,0.7) and f(0,0.3). This advanced problem covers key mathematical concepts like Gaussian distributions, variance addition, and Fourier transforms. Ideal for university-level students studying probability and signal processing.

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