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Marginal distribution
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Part of a series on Statistics
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Probability axioms
Probability space
Sample space
Elementary event
Event
Random variable
Probability measure
Complementary event
Joint probability
Marginal probability
Conditional probability
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In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables. This contrasts with a conditional distribution, which gives the probabilities contingent upon the values of the other variables.
Marginal variables are those variables in the subset of variables being retained. These concepts are "marginal" because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table.[1] The distribution of the marginal variables (the marginal distribution) is obtained by marginalizing – that is, focusing on the sums in the margin – over the distribution of the variables being discarded, and the discarded variables are said to have been marginalized out.
The context here is that the theoretical studies being undertaken, or the data analysis being done, involves a wider set of random variables but that attention is being limited to a reduced number of those variables. In many applications, an analysis may start with a given collection of random variables, then first extend the set by defining new ones (such as the sum of the original random variables) and finally reduce the number by placing interest in the marginal distribution of a subset (such as the sum). Several different analyses may be done, each treating a different subset of variables as the marginal variables.
Contents
[hide]
1 Definition
1.1 Two-variable Case
1.1.1 Marginal Probability Mass and Density Functions
1.1.2 Marginal Probability
2 Real-world example
3 Multivariate distributions
4 See also
5 References
6 Bibliography
Definition[edit]
Two-variable Case[edit]
Given two random variables X and Y whose joint distribution is known, the marginal distribution of X is simply the probability distribution of X averaging over information about Y. It is the probability distribution of X when the value of Y is not known. This is typically calculated by summing or integrating the joint probability distribution over Y.
X
Y
x1
x2
x3
x4
py(Y)↓
y1
4⁄32 2⁄32 1⁄32 1⁄32
8⁄32
y2
2⁄32 4⁄32 1⁄32 1⁄32
8⁄32
y3
2⁄32 2⁄32 2⁄32 2⁄32
8⁄32
y4
8⁄32 0 0 0
8⁄32
px(X) →
16⁄32
8⁄32
4⁄32
4⁄32
32⁄32
Joint and marginal distributions of a pair of discrete, random variables X,Y having nonzero mutual information I(X; Y). The values of the joint distribution are in the 4×4 square, and the values of the marginal distributions are along the right and bottom margins.
Marginal Probability Mass and Density Functions[edit]
For discrete random variables, the marginal probability mass function can be written as Pr(X = x). This is
Pr ( X = x ) = ∑ y Pr ( X = x , Y = y ) = ∑ y Pr ( X = x ∣ Y = y ) Pr ( Y = y ) , {\displaystyle \Pr(X=x)=\sum _{y}\Pr(X=x,Y=y)=\sum _{y}\Pr(X=x\mid Y=y)\Pr(Y=y),}
where Pr(X = x,Y = y) is the joint distribution of X and Y, while Pr(X = x|Y = y) is the conditional distribution of X given Y. In this case, the variable Y has been "marginalized out".
Bivariate marginal and joint probabilities for discrete random variables are often displayed as two-way tables.
Similarly for continuous random variables, the marginal probability density function can be written as pX(x). This is
p X ( x ) = ∫ y p X , Y ( x , y ) d y = ∫ y p X ∣ Y ( x ∣ y ) p Y ( y ) d y , {\displaystyle p_{X}(x)=\int _{y}p_{X,Y}(x,y)\,\mathrm {d} y=\int _{y}p_{X\mid Y}(x\mid y)\,p_{Y}(y)\,\mathrm {d} y,}
where pX,Y(x,y) gives the joint distribution of X and Y, while pX|Y(x|y) gives the conditional distribution for X given Y. Again, the variable Y has been "marginalized out".
Marginal Probability[edit]
A marginal probability can always be written as an expected value:
p X ( x ) = ∫ y p X ∣ Y ( x ∣ y ) p Y ( y ) d y = E Y [ p X ∣ Y ( x ∣ y ) ] {\displaystyle p_{X}(x)=\int _{y}p_{X\mid Y}(x\mid y)\,p_{Y}(y)\,\mathrm {d} y=\mathbb {E} _{Y}[p_{X\mid Y}(x\mid y)]}
Intuitively, the marginal probability of X is computed by examining the conditional probability of X given a particular value of Y, and then averaging this conditional probability over the distribution of all values of Y.
This follows from the definition of expected value (after applying the law of the unconscious statistician):
E Y [ f ( Y ) ] = ∫ y f ( y ) p Y ( y ) d y . {\displaystyle \mathbb {E} _{Y}[f(Y)]=\int _{y}f(y)p_{Y}(y)\,\mathrm {d} y.}
Therefore marginalization provides the rule for the transformation of the probability distribution of a random variable Y and another random variable X = g(Y):
p X ( x ) = ∫ y p X ∣ Y ( x ∣ y ) p Y ( y ) d y = ∫ y δ ( x − g ( y ) ) p Y ( y ) d y , {\displaystyle p_{X}(x)=\int _{y}p_{X\mid Y}(x\mid y)\,p_{Y}(y)\,\mathrm {d} y=\int _{y}\delta (x-g(y))\,p_{Y}(y)\,\mathrm {d} y,}
Real-world example[edit]
Suppose that the probability that a pedestrian will be hit by a car, while crossing the road at a pedestrian crossing, without paying attention to the traffic light, is to be computed. Let H be a discrete random variable taking one value from {Hit, Not Hit}. Let L (for traffic light) be a discrete random variable taking one value from {Red, Yellow, Green}.
Realistically, H will be dependent on L. That is, P(H = Hit) will take different values depending on whether L is red, yellow or green (and likewise for P(H = Not Hit)). A person is, for example, far more likely to be hit by a car when trying to cross while the lights for perpendicular traffic are green than if they are red. In other words, for any given possible pair of values for H and L, one must consider the joint probability distribution of H and L to find the probability of that pair of events occurring together if the pedestrian ignores the state of the light.
However, in trying to calculate the marginal probability P(H=hit), what we are asking for is the probability that H=Hit in the situation in which we don't actually know the particular value of L and in which the pedestrian ignores the state of the light. In general, a pedestrian can be hit if the lights are red OR if the lights are yellow OR if the lights are green. So, the answer for the marginal probability can be found by summing P(H | L) for all possible values of L, with each value of L weighted by its probability of occurring.
Here is a table showing the conditional probabilities of being hit, depending on the state of the lights. (Note that the columns in this table must add up to 1 because the probability of being hit or not hit is 1 regardless of the state of the light.)
Conditional distribution: P ( H | L ) {\displaystyle P(H|L)}
L
H
Red
Yellow
Green
Not Hit
0.99 0.9 0.2
Hit
0.01 0.1 0.8
To find the joint probability distribution, we need more data. Let's say that P(L=red) = 0.2, P(L=yellow) = 0.1, and P(L=green) = 0.7. Multiplying each column in the conditional distribution by the probability of that column occurring, we find the joint probability distribution of H and L, given in the central 2×3 block of entries. (Note that the cells in this 2×3 block add up to 1).
Joint distribution: P ( H , L ) {\displaystyle P(H,L)}
L
H
Red
Yellow
Green
Marginal probability P(H)
Not Hit
0.198 0.09 0.14 0.428
Hit
0.002 0.01 0.56 0.572
Total
0.2 0.1 0.7 1
The marginal probability P(H=Hit) is the sum 0.572 along the H=Hit row of this joint distribution table, as this is the probability of being hit when the lights are red OR yellow OR green. Similarly, the marginal probability that P(H=Not Hit) is the sum along the H=Not Hit row.
Multivariate distributions[edit]
Many samples from a bivariate normal distribution. The marginal distributions are shown in red and blue. The marginal distribution of X is also approximated by creating a histogram of the X coordinates without consideration of the Y coordinates.
For multivariate distributions, formulae similar to those above apply with the symbols X and/or Y being interpreted as vectors. In particular, each summation or integration would be over all variables except those contained in X.
See also[edit]
Joint probability distribution
Compound probability distribution
Wasserstein metric
References[edit]
Jump up ^ Trumpler, Robert J. and Harold F. Weaver (1962). Statistical Astronomy. Dover Publications. pp. 32–33.
Bibliography[edit]
Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press. ISBN 0-521-81099-X.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Marginal_distribution&oldid=841154304"
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Channel Ports
From Wikipedia, the free encyclopedia
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The Channel Ports are seaports in southern England and the facing continent, which allow for short crossings of the English Channel. There is no formal definition, but there is a general understanding of the term. Some ferry companies divide their routes into "short" and "long" crossings. The broadest definition might be from Plymouth east to Kent and from Roscoff to Zeebrugge although a tighter definition would exclude ports west of Newhaven and Dieppe. A historic group of such ports is the Cinque Ports of south-east England, most of which have ceased to be commercial ports.[1]
Contents
1 Ports
1.1 England
1.2 France
2 Strategic importance
2.1 World War I
2.2 World War II
3 See also
4 References
4.1 Notes
4.2 Bibliography
5 External links
Ports[edit]
England[edit]
Aerial view of Dover harbour
The ports vary in size and their relative importance has fluctuated during recent history. Dover has established a lead in the cross-Channel ferry routes through its geographic position and development of its facilities and hinterland. This business has been sustained despite competition from the Channel Tunnel. Other minor ports in Kent and Sussex have retained some trade but these tend to be single routes, such as Folkestone–Calais, Newhaven–Dieppe and Ramsgate–Ostend.
Longer routes mainly radiate from Portsmouth but there are lesser ports at Poole, Portland and Plymouth, with routes to Normandy, Brittany and Spain.
France[edit]
The major French port with cross-Channel connections is Calais, principally with Dover but also with Folkestone. Other close French channel include Dunkirk, Boulogne and Dieppe.
Longer routes are served from Le Havre, Caen, Cherbourg, Roscoff and St Malo. There are services to Ireland (Rosslare and Cork).
Strategic importance[edit]
The ports are important commercial facilities, reinforcing connections between the British and European road systems. They are also vulnerable to industrial action such as strikes by port employees[2] or blockades by disgruntled fishermen.[3]
Their importance as military facilities was demonstrated during two World Wars.
World War I[edit]
During World War I the British and French Channel ports were major conduits for British materiel and troops.
The Belgian ports of Ostend and Zeebrugge were considered a major threat by British Admiral Admiral Jellicoe. He was concerned by their use not only as German U-boat ports, but also as torpedo boat bases and even possible departure points for a cross-Channel attack. This concern was transmitted via Whitehall to the British chief of staff on the Western Front, General Haig, for whom it merely confirmed the need for an offensive in Flanders, and eventually led to the Battle of Passchendaele (also known as the Third Battle of Ypres).[4]
World War II[edit]
During World War II, likewise, the Channel ports provided major supply routes which had to be reopened in 1944. Dunkirk was the route from which British and Allied troops were evacuated in 1940; see Dunkirk evacuation
In September 1944, the First Canadian Army (Canadian, British, Polish, Czechoslovak and other national units) was to capture the ports from Le Havre to Zeebrugge. Dieppe and Ostende were undefended but Le Havre, Boulogne and Calais required major military actions. It took several weeks to bring the ports back into use at a time when Allied armies badly needed supplies. Dunkirk was left under siege until the general German surrender.[5]
See also[edit]
Operation Stack
List of ports and harbours of the Atlantic Ocean
Cinque Ports (in England)
References[edit]
Notes[edit]
Jump up ^ "Channel Ferry Ports linking England and France". visitFrance. 2002–2009. Retrieved 15 July 2009.
Jump up ^ "Dover-Calais/Boulogne ferry services hit by strike". eTravel.org. 16 October 2008. Archived from the original on 14 April 2013. Retrieved 15 July 2009.
Jump up ^ "Ports blocked in cod quota protest". CNN. 11 December 2002. Retrieved 15 July 2009.
Jump up ^ Strachan, Hew. The First World War. Pocket Books, 2006, pp. 244–5.
Jump up ^ Stacey. "Clearing the Coastal Belt and the Ports, September 1944". Department of National Defence. Retrieved 15 July 2009.
Bibliography[edit]
Cowsill, Miles; Hendy, John (1994). Newhaven-Dieppe: the car ferry era. Kilgetty, Pembrokeshire: Ferry Publications. ISBN 1871947200.
Cowsill, Miles (1997). Ferries of The English Channel: Past and Present. Kilgetty, Pembrokeshire: Ferry Publications. ISBN 187194743X.
Hendy, John (1991). The Dover–Ostend Line. Staplehurst, Kent: Ferry Publications. ISBN 095135065X.
Hendy, John (1993). Ferries of Dover. Staplehurst, Kent: Ferry Publications. ISBN 0951350692.
Hendy, John (1997). Ferry Port Dover: the development of cross-channel vehicle ferries, their services and allied infrastructure. Staplehurst, Kent: Ferry Publications. ISBN 1871947472.
Hendy, John (2009). Dover-Calais: The Short-Sea Route. Ramsey, Isle of Man: Ferry Publications. ISBN 9781871947939.
Hendy, John (2014). Folkestone: for the Continent. Ramsey, Isle of Man: Ferry Publications. ISBN 9781906608767.
Hendy, John (2016). Dover-Calais. Ramsey, Isle of Man: Ferry Publications. ISBN 9781906608743.
Pattheeuws, Stephen (2015). The Ostend Ferry: from start to finish. Ramsey, Isle of Man: Ferry Publications. ISBN 9781906608804.
Searle, Brian; Peter, Bruce (2010). Weymouth Ferries: The Rise and Fall of a Port. Ramsey, Isle of Man: Ferry Publications. ISBN 9781906608088.
External links[edit]
A commercial site promoting cross-channel traffic
List of ferry routes to England
List of ferry routes to France
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