Question

1. Choose the statement below that is not true:

A: The Empirical rule applies to all normal distributions.

B: Both the Poisson and the exponential distributions are characterized by only their mean.

C: All of the statements are true.

D: Poisson distributions tend to be less useful the higher the mean is.

E: The exponential distribution tends to be a good fit for counting the discrete number of events per time interval.

2.Combining the standard deviations of two negatively correlated normally distributed random variables results in a lower combined standard deviation than if the same two standard deviations were uncorrelated. (Hint: come up with an example and test against it)

True

False

Answer 1

1.

the statement below that is not true: :

E: The exponential distribution tends to be a good fit for counting the discrete number of events per time interval.

Explanation:

A: The Empirical rule applies to all normal distributions.

True

B: Both the Poisson and the exponential distributions are characterized by only their mean.

True

D: Poisson distributions tend to be less useful the higher the mean is.

True

because if the mean is large, Poisson Distribution approximates a normal distribution.

E: The exponential distribution tends to be a good fit for counting the discrete number of events per time interval.

False

because Poisson Distribution deals with the number of occurrences in a fixed period of time. Exponential distribution deals with the time between occurrences of successive events as time flows continuously.

2.

Correct option:

True

Explanation:

By Theorem:

Var (X + Y) = Var (X) + Var (Y) + 2 Cov (X,Y)

So,for example:

Consider two uncorrelated variables X and Y. So,by the above Theorem, we have

Var (X + Y) = Var (X) + Var (Y)

Now consider two negatively correlated variables X and Y.

By the property of Covariance, we have: Cov (X,Y) will be negative.

Thus,

Substituting in the above Theorem, we prove the required result:

Combining the standard deviations of two negatively correlated normally distributed random variables results in a lower combined standard deviation than if the same two standard deviations were uncorrelated.