Question:
Consider the normal probability distribution function
\[
f(x) = \frac{4}{\sqrt{2\pi}} e^{-8(x+3)^2}.
\]
If \( \mu \) and \( \sigma \) are the mean and standard deviation of \( f(x) \) respectively, then the ordered pair \( (\mu, \sigma) \) is:
Consider the normal probability distribution function
\[
f(x) = \frac{4}{\sqrt{2\pi}} e^{-8(x+3)^2}.
\]
If \( \mu \) and \( \sigma \) are the mean and standard deviation of \( f(x) \) respectively, then the ordered pair \( (\mu, \sigma) \) is:
Show Hint
In normal distribution problems, identify the mean (\( \mu \)) from the shift in \( x \), and calculate the standard deviation (\( \sigma \)) using the coefficient of \( x^2 \) in the exponential.
Updated On: Jan 24, 2025
- \( \left( 3, \frac{1}{4} \right) \)
- \( \left( -3, \frac{1}{4} \right) \)
- \( \left( 3, 4 \right) \)
- \( \left( -3, 4 \right) \)
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The Correct Option is B
Solution and Explanation
Step 1: Analyze the general form of the normal distribution.
The general form of the normal probability distribution is:
\[
f(x) = \frac{1}{\sqrt{2\pi} \sigma} e^{-\frac{(x - \mu)^2}{2\sigma^2}},
\]
where \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
Step 2: Match the given equation to the general form.
The given function is:
\[
f(x) = \frac{4}{\sqrt{2\pi}} e^{-8(x+3)^2}.
\]
Compare this with the general form. The coefficient of \( (x + 3)^2 \) inside the exponent is \( 8 \). This gives:
\[
\frac{1}{2\sigma^2} = 8 \quad \Rightarrow \quad \sigma^2 = \frac{1}{16} \quad \Rightarrow \quad \sigma = \frac{1}{4}.
\]
The mean \( \mu \) is obtained from the shift \( (x + 3) \), which indicates \( \mu = -3 \).
Step 3: Conclusion.
The ordered pair \( (\mu, \sigma) \) is \( \left( -3, \frac{1}{4} \right) \).
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