Normal distribution var(x)=sigma^2

In summary, "var(x)=sigma^2" in a normal distribution refers to the variance of a random variable x, which represents the average squared deviation from the mean. The value of sigma, or the standard deviation, affects the shape of the normal distribution curve. The mean, median, and mode are all equal and located at the center of a normal distribution. The "68-95-99.7 rule" is related to the value of sigma in the equation "var(x)=sigma^2". This equation cannot be used to analyze non-normal data.
  • #1
badgerbadger
7
0
suppose that X is normally distributed with mean u and standard deviation sigma. show that var(x)=sigma^2.[you many use the fact that if Z is standard normally distributed, then EZ=0 and var(x)=1]
 
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  • #2
Okay, what is the definition of "variance"? What is the definition of "standard deviation"? The result should be immediate from the definitions.
 

1. What is the meaning of "var(x)=sigma^2" in normal distribution?

The term "var(x)=sigma^2" refers to the variance of a random variable x in a normal distribution. It represents the average squared deviation from the mean of the distribution.

2. How does the value of sigma affect the shape of the normal distribution curve?

The value of sigma, also known as the standard deviation, determines the spread or width of the normal distribution curve. A larger value of sigma results in a wider and flatter curve, while a smaller value of sigma produces a narrower and taller curve.

3. What is the relationship between the mean, median, and mode in a normal distribution?

In a normal distribution, the mean, median, and mode are all equal and located at the center of the distribution. This is because the normal distribution is symmetric around its mean.

4. How is the concept of "68-95-99.7 rule" related to normal distribution var(x)=sigma^2?

The "68-95-99.7 rule" states that in a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, about 95% falls within two standard deviations, and nearly all (99.7%) falls within three standard deviations. This rule is directly related to the value of sigma, which is the standard deviation, in the equation "var(x)=sigma^2".

5. Can the normal distribution var(x)=sigma^2 be used to analyze non-normal data?

No, the normal distribution var(x)=sigma^2 is specifically designed for analyzing data that follows a normal distribution. It cannot be used to analyze non-normal data as it will not accurately represent the distribution of the data.

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