Random Variables an important concept in probability

A random variable, X, is a numerical quantity whose value is determined be a random experiment Examples 1.Two dice are rolled and X is the sum of the two upward faces. 2.A coin is tossed n = 3 times and X is the number of times that a head occurs. 3.We count the number of earthquakes, X, that occur in the San Francisco region from 2000 A. D, to 2050A. D. 4.Today the TSX composite index is 11,050.00, X is the value of the index in thirty days

Examples – R.V.’s - continued 5.A point is selected at random from a square whose sides are of length 1. X is the distance of the point from the lower left hand corner. 6.A chord is selected at random from a circle. X is the length of the chord. point X chord X

Definition – The probability function, p(x), of a random variable, X. For any random variable, X, and any real number, x, we define where {X = x} = the set of all outcomes (event) with X = x.

Definition – The cumulative distribution function, F(x), of a random variable, X. For any random variable, X, and any real number, x, we define where {X ≤ x} = the set of all outcomes (event) with X ≤ x.

(1,1) 2 (1,2) 3 (1,3) 4 (1,4) 5 (1,5) 6 (1,6) 7 (2,1) 3 (2,2) 4 (2,3) 5 (2,4) 6 (2,5) 7 (2,6) 8 (3,1) 4 (3,2) 5 (3,3) 6 (3,4) 7 (3,5) 8 (3,6) 9 (4,1) 5 (4,2) 6 (4,3) 7 (4,4) 8 (4,5) 9 (4,6) 10 (5,1) 6 (5,2) 7 (5,3) 8 (5,4) 9 (5,5) 10 (5,6) 11 (6,1) 7 (6,2) 8 (6,3) 9 (6,4) 10 (6,5) 11 (6,6) 12 Examples 1.Two dice are rolled and X is the sum of the two upward faces. S, sample space is shown below with the value of X for each outcome

Graph x p(x)p(x)

The cumulative distribution function, F(x) For any random variable, X, and any real number, x, we define where {X ≤ x} = the set of all outcomes (event) with X ≤ x. Note {X ≤ x} =  if x < 2. Thus F(x) = 0. {X ≤ x} =  {(1,1)} if 2 ≤ x < 3. Thus F(x) = 1/36 { X ≤ x} =  {(1,1),(1,2),(1,2)} if 3 ≤ x < 4. Thus F(x) = 3/36

Continuing we find F(x) is a step function

2.A coin is tossed n = 3 times and X is the number of times that a head occurs. The sample Space S = {HHH (3), HHT (2), HTH (2), THH (2), HTT (1), THT (1), TTH (1), TTT (0)} for each outcome X is shown in brackets

Graph probability function p(x)p(x) x

Graph Cumulative distribution function F(x)F(x) x

Examples – R.V.’s - continued 5.A point is selected at random from a square whose sides are of length 1. X is the distance of the point from the lower left hand corner. 6.A chord is selected at random from a circle. X is the length of the chord. point X chord X

Examples – R.V.’s - continued 5.A point is selected at random from a square whose sides are of length 1. X is the distance of the point from the lower left hand corner. An event, E, is any subset of the square, S. P[E] = (area of E)/(Area of S) = area of E point X S E

Thus p(x) = 0 for all values of x. The probability function for this example is not very informative S The probability function

The Cumulative distribution function S

S

The probability density function, f(x), of a continuous random variable Suppose that X is a random variable. Let f(x) denote a function define for -∞ < x < ∞ with the following properties: 1. f(x) ≥ 0 Then f(x) is called the probability density function of X. The random, X, is called continuous.

Probability density function, f(x)

Cumulative distribution function, F(x)

Thus if X is a continuous random variable with probability density function, f(x) then the cumulative distribution function of X is given by: Also because of the fundamental theorem of calculus.

Example A point is selected at random from a square whose sides are of length 1. X is the distance of the point from the lower left hand corner. point X

Now

Also

Now and

Finally

Graph of f(x)

Summary

Discrete random variables For a discrete random variable X the probability distribution is described by the probability function, p(x), which has the following properties : This denotes the sum over all values of x between a and b.

Graph: Discrete Random Variable p(x)p(x) a b

Continuous random variables For a continuous random variable X the probability distribution is described by the probability density function f(x), which has the following properties : 1. f(x) ≥ 0

Graph: Continuous Random Variable probability density function, f(x)

A Probability distribution is similar to a distribution of mass. A Discrete distribution is similar to a point distribution of mass. Positive amounts of mass are put at discrete points. x1x1 x2x2 x3x3 x4x4 p(x1)p(x1) p(x2)p(x2) p(x3)p(x3) p(x4)p(x4)

A Continuous distribution is similar to a continuous distribution of mass. The total mass of 1 is spread over a continuum. The mass assigned to any point is zero but has a non-zero density f(x)f(x)

The distribution function F(x) This is defined for any random variable, X. F(x) = P[X ≤ x] Properties 1. F(-∞) = 0 and F(∞) = 1. Since {X ≤ - ∞} =  and {X ≤ ∞} = S then F(- ∞) = 0 and F(∞) = 1.

2. F(x) is non-decreasing (i. e. if x 1 < x 2 then F(x 1 ) ≤ F(x 2 ) ) 3. F(b) – F(a) = P[a < X ≤ b]. If x 1 < x 2 then {X ≤ x 2 } = {X ≤ x 1 }  {x 1 < X ≤ x 2 } Thus P[X ≤ x 2 ] = P[X ≤ x 1 ] + P[x 1 < X ≤ x 2 ] or F(x 2 ) = F(x 1 ) + P[x 1 < X ≤ x 2 ] Since P[x 1 < X ≤ x 2 ] ≥ 0 then F(x 2 ) ≥ F(x 1 ). If a < b then using the argument above F(b) = F(a) + P[a < X ≤ b] Thus F(b) – F(a) = P[a < X ≤ b].

4. p(x) = P[X = x] =F(x) – F(x-) 5.If p(x) = 0 for all x (i.e. X is continuous) then F(x) is continuous. Here A function F is continuous if One can show that Thus p(x) = 0 implies that

For Discrete Random Variables F(x) is a non-decreasing step function with F(x)F(x) p(x)p(x)

For Continuous Random Variables Variables F(x) is a non-decreasing continuous function with F(x)F(x) f(x) slope x