Normally distributed and uncorrelated does not imply independent and Midge Hall railway station

In probability theory, two random variables being uncorrelated does not imply their independence. In some contexts, uncorrelatedness implies at least pairwise independence (as when the random variables involved have Bernoulli distributions).

It is sometimes mistakenly thought that one context in which uncorrelatedness implies independence is when the random variables involved are normally distributed. However, this is incorrect if the variables are merely marginally normally distributed but not jointly normally distributed.

Suppose two random variables X and Y are jointly normally distributed. That is the same as saying that the random vector (X, Y) has a multivariate normal distribution. It means that the joint probability distribution of X and Y is such that each linear combination of X and Y is normally distributed, i.e. for any two constant (i.e., non-random) scalars a and b, the random variable aX + bY is normally distributed. In that case if X and Y are uncorrelated, i.e., their covariance cov(X, Y) is zero, then they are independent. However, it is possible for two random variables X and Y to be so distributed jointly that each one alone is marginally normally distributed, and they are uncorrelated, but they are not independent; examples are given below.

Contents 1 Examples 1.1 A symmetric example 1.2 An asymmetric example 2 References

Examples A symmetric example Joint range of X and Y. Darker indicates higher value of the density function.

Suppose X has a normal distribution with expected value 0 and variance 1. Let W have the Rademacher distribution, so that W = 1 or −1, each with probability 1/2, and assume W is independent of X. Let Y = WX. Then X and Y are uncorrelated; Both have the same normal distribution; and X and Y are not independent.

Note that the distribution of the simple linear combination X + Y concentrates positive probability at 0: Pr(X + Y = 0)  = 1/2 and so is not normally distributed. By the definition above, X and Y are not jointly normally distributed.

To see that X and Y are uncorrelated, consider

To see that Y has the same normal distribution as X, consider

(since X and −X both have the same normal distribution), where is the cumulative distribution function of the normal distribution..

To see that X and Y are not independent, observe that |Y| = |X| or that Pr(Y > 1 | X = 1/2) = Pr(X > 1 | X = 1/2) = 0. An asymmetric example The joint density of X and Y. Darker indicates a higher value of the density.

Suppose X has a normal distribution with expected value 0 and variance 1. Let

where c is a positive number to be specified below. If c is very small, then the correlation corr(X, Y) is near −1; if c is very large, then corr(X, Y) is near 1. Since the correlation is a continuous function of c, the intermediate value theorem implies there is some particular value of c that makes the correlation 0. That value is approximately 1.54. In that case, X and Y are uncorrelated, but they are clearly not independent, since X completely determines Y.

To see that Y is normally distributed—indeed, that its distribution is the same as that of X—let us find its cumulative distribution function:

where the next-to-last equality follows from the symmetry of the distribution of X and the symmetry of the condition that |X| ≤ c.

Observe that the difference X − Y is nowhere near being normally distributed, since it has a substantial probability (about 0.88) of it being equal to 0, whereas the normal distribution, being a continuous distribution, has no discrete part, i.e., does not concentrate more than zero probability at any single point. Consequently X and Y are not jointly normally distributed, even though they are separately normally distributed.

Midge Hall railway station and Normally distributed and uncorrelated does not imply independent

Midge Hall railway station was located in Midge Hall, Leyland, closing to passengers in 1961, although the line still exists as the Ormskirk Branch Line. History

The railway line between Preston and Walton was proposed by the Liverpool, Ormskirk and Preston Railway (LO&PJ) and authorised in 1846; later that year the LO&PJ was amalgamated with the East Lancashire Railway (ELR), which opened the line in 1849.

In August 1859 the ELR was amalgamated with the Lancashire and Yorkshire Railway (LYR), and in October that year, the station at Midge Hall was opened. It was 23 1⁄4 miles (37.4 km) from Liverpool (Tithebarn Street), and replaced an earlier station at Cocker Bar, 23 miles (37.0 km) from Liverpool.

The station was closed by British Railways on 2 October 1961. It retained its original Lancashire and Yorkshire railway signalbox until 1972 until the general Preston area resignalling programme, whereupon the old box was demolished and replaced with a new construction on the opposite side of the level crossing.

Trains still stop at Midge Hall signal box to exchange a token for the single line onward to Rufford- this is a vestige of the 1970s and early 1980s, when the then recently singled branchline retained double track from Midge Hall into Preston. Reopening proposals

There have been talks amongst the local community for the possible reopening of the station. A study held in 1991 concluded that there would be a forecasted 7500 journeys per annum using the station, generating roughly £15,000 in revenue with an average cost of £2 per journey. This was deemed uneconomical due to the high costs of construction (£500,000) coupled with £15,000 in ongoing annual maintenance costs, which would barely be covered by the revenue alone.

A meeting held in 2003 concluded that whilst forecasted passenger numbers will likely be higher than that of the 1991 study (due largely to residential development in recent years), numbers may only be in the region of 10,000 to 30,000 and revenue from these passengers would likely still not cover the costs for reopening and thus remains uneconomical to reopen.

In 2012, the Ormskirk, Preston and Southport Travellers’ Association called talks with Lancashire County Council about the possible reopening of the Midge Hall station as a "key component" of the Council's thinking. In the summer of 2014, Lancashire County Council confirmed that a business case for reopening was being formulated.
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