In a normal distribution, increasing the expected value shifts the curve horizontally but does not change its shape.

In a normal distribution, increasing the expected value shifts the curve horizontally but does not change its shape.

• A. True
• B. False
• C. It depends on the variance
• D. Only for standardized values

Answer: (A)

In a normal distribution, the mean shifts where the peak sits, while the standard deviation controls how wide or narrow the bell is. The probability density function is f(x) = (1/(sigma sqrt(2pi))) exp(- (x - mu)^2 /(2 sigma^2)). Here, mu is the location parameter: increasing mu moves the entire curve to the right along the x-axis, but the height and width are governed by sigma. Since sigma stays the same, the shape—the bell’s proportions—stays unchanged; only its position changes. If you were to change the variance (sigma), the shape would change, but changing the mean alone just translates the curve. So the statement is true.

In a normal distribution, the mean shifts where the peak sits, while the standard deviation controls how wide or narrow the bell is. The probability density function is f(x) = (1/(sigma sqrt(2pi))) exp(- (x - mu)^2 /(2 sigma^2)). Here, mu is the location parameter: increasing mu moves the entire curve to the right along the x-axis, but the height and width are governed by sigma. Since sigma stays the same, the shape—the bell’s proportions—stays unchanged; only its position changes. If you were to change the variance (sigma), the shape would change, but changing the mean alone just translates the curve. So the statement is true.