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Probability · Lecture 40 of 76 · 6:47

Recitation: Ambulance Travel Time

Study guide

What this lecture covers

This short problem finds the distribution of the time it takes an ambulance to travel from its current position to an accident, when both positions are independent and uniformly distributed along a stretch of road. It is a geometric derived-distribution problem: the travel time is a function of two random variables rather than one.

It applies the CDF-then-differentiate method from the derived-distributions lecture to a case involving an absolute difference of two independent uniform random variables, translating the probability into an area calculation.

Key ideas

Travel time as a function of two variables: with accident location X and ambulance location Y both uniform on [0,l] and independent, travel time is T = |Y - X| / v for a constant speed v.
Translating the CDF into a region: P(T <= t) becomes the event that Y lies within vt of X, a band-shaped region in the (x,y) square.
Probability as area under a uniform joint density: since X and Y are independent and uniform, the joint density is a constant, so the probability equals that constant times the area of the band inside the square.
Area by subtraction: the band's area is found by subtracting the area of two corner triangles (where the band runs off the edge of the square) from the full square.
Differentiating the CDF piecewise: the resulting CDF is piecewise, and differentiating it in the middle region gives a PDF, while the value is constant (and so has zero density) outside that range.

Walkthrough

Setting up T as a function of X and Y (0:00)

The accident location X and the ambulance's location Y are each uniform on [0,l] and independent. Travel time is T = |Y-X| / v. The event T <= t is rewritten as X - vt <= Y <= X + vt, defining a band-shaped region A in the square [0,l] x [0,l].

Computing the probability as an area (3:12)

Since X and Y are independent and uniform, their joint density is the constant 1/l^2. The probability that T <= t is this constant times the area of region A, which is found by subtracting two triangular corner regions (each with area (1/2)(l - vt)^2) from the full square l^2.

Differentiating to get the PDF (5:16)

Differentiating the resulting CDF with respect to t, for t between 0 and l/v, gives a PDF of 2v/l - 2v^2 t / l^2. Outside that range, the CDF is constant (0 below the range, 1 above it), so the density is 0 there.

Before you watch

Be comfortable with the CDF-then-differentiate method for finding the distribution of a function of random variables.
Reviewing how probabilities correspond to areas under a uniform joint density on a square will help with the geometric argument here.

Check your understanding

Why is the event T <= t equivalent to Y falling within vt of X?
Why does the joint density of X and Y being constant let you replace an integral with an area calculation?
Why is the PDF of T equal to 0 for t greater than l/v?

Vocabulary

constant speed (phrase): A speed that does not change over time.
The ambulance travels at a constant speed v.
band-shaped region (phrase): An area shaped like a straight strip between two parallel lines.
The event forms a band-shaped region in the square.
corner triangle (phrase): A triangular area located at the corner of a larger shape.
We subtract two corner triangles to find the band's area.
subtraction (noun): The act of taking one amount away from another.
The area is found by subtraction of the triangles from the square.
differentiate (verb): To calculate the rate of change of a function.
We differentiate the CDF to get the PDF.
stretch of road (phrase): A continuous section of a road.
Both locations are uniform along a stretch of road.
CDF (phrase): A function giving the probability that a random variable is less than or equal to a value.
We find the CDF of the travel time T.
PDF (phrase): A curve that describes how likely different values of a continuous variable are.
Differentiating the CDF gives the PDF of T.
joint density (phrase): A function describing how likely combinations of two or more variables are together.
The joint density of X and Y is constant over the square.
uniform (adjective): Equally likely across a whole range of values.
Both X and Y are uniform on [0, l].
translate (verb): To express one idea or quantity in terms of another.
We translate the CDF into an area calculation.
region (noun): A defined area within a larger space.
The band A is a region inside the square.
piecewise (adjective): Defined by different rules in different parts of its range.
The resulting CDF is piecewise, with different behavior outside the middle range.
area (noun): The amount of space inside a two-dimensional shape.
The probability equals a constant times the area of the band.
geometric argument (phrase): A reasoning method based on shapes and areas rather than direct calculation.
A geometric argument turns the probability into an area problem.
derived distribution (phrase): The probability distribution of a new variable built as a function of other random variables.
This is a derived-distribution problem for a function of two variables.
absolute difference (phrase): The distance between two numbers, always positive or zero.
Travel time depends on the absolute difference between X and Y.
edge (noun): The boundary line of a shape.
The band runs off the edge of the square near the corners.
apply (verb): To use a rule or method in a specific situation.
We apply the CDF-then-differentiate method here.
function (noun): A rule that produces one output value from one or more input values.
Travel time T is a function of X and Y.
event (noun): A specific outcome or set of outcomes being considered.
The event T <= t corresponds to a region in the square.
location (noun): A specific position or point in space.
The accident location X is uniform along the road.
square (noun): A shape with four equal sides and four right angles.
The region A lies inside the square [0,l] x [0,l].
rewritten (adjective): Expressed again in a different, equivalent form.
The event is rewritten as X - vt <= Y <= X + vt.
span (verb): To extend or reach across a range.
The band spans a strip around the diagonal of the square.

From the YouTube description

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
View the complete course: http://ocw.mit.edu/6-041SCF13
Instructor: Kuang Xu

License: Creative Commons BY-NC-SA
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