MA 320-001: Introductory Probability David Murrugarra Department of Mathematics, University of Kentucky http://www.math.uky.edu/~dmu228/ma320/ Spring 2017 David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 1 / 12
Suppose X and Y are two independent discrete random variables with distribution functions f 1 (x) and f 2 (x). Let Z = X + Y. David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 2 / 12
Suppose X and Y are two independent discrete random variables with distribution functions f 1 (x) and f 2 (x). Let Z = X + Y. Suppose that X = k, where k is some integer. Then Z = z if and only if Y = z k. So the event Z = z is the union of the pairwise disjoint events {X = k} and {Y = z k} where k runs over the integers. David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 2 / 12
Suppose X and Y are two independent discrete random variables with distribution functions f 1 (x) and f 2 (x). Let Z = X + Y. Suppose that X = k, where k is some integer. Then Z = z if and only if Y = z k. So the event Z = z is the union of the pairwise disjoint events {X = k} and {Y = z k} where k runs over the integers. Since these events are pairwise disjoint, we have P(Z = z) = k= P(X = k) P(Y = z k) Thus, we have found the distribution function of the random variable Z. This leads to the following definition. David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 2 / 12
Definition (Convolutions) Let X and Y be two independent integer-valued random variables, with density functions f 1 (x) and f 2 (x) respectively. Then the convolution of f 1 (x) and f 2 (x) is the distribution function f 3 = f 1 f 2 given by f 3 (j) = (f 1 f 2 )(j) = k f 1 (k) f 2 (j k) for j =..., 2, 1, 0, 1, 2,... The function f 3 (x) is the distribution function of the random variable Z = X + Y. It is easy to see that the convolution operation is commutative, and it is straightforward to show that it is also associative. David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 3 / 12
Let S n = X 1 + X 2 + + X n be the sum of n independent random variables of an independent trials process with common distribution function f defined on the integers. David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 4 / 12
Let S n = X 1 + X 2 + + X n be the sum of n independent random variables of an independent trials process with common distribution function f defined on the integers. Then the distribution function of S 1 is f. David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 4 / 12
Let S n = X 1 + X 2 + + X n be the sum of n independent random variables of an independent trials process with common distribution function f defined on the integers. Then the distribution function of S 1 is f. And we can write S n = S n 1 + X n Thus, since we know the distribution function of X n is f, we can find the distribution function of S n by induction. David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 4 / 12
Example (Homework12: Problem 1) Let X and Y be independent discrete random variables: X can be wither 1, 2, or 3, while Y can be either 4, 5, or 6. They have the following probability distributions: (a) Find P(X = 1 and Y = 4). (b) Find P(X + Y = 5). (c) Find P(X + Y = 6). (d) Find P(X + Y = 7). x P(X = x) y P(Y = y) 1 0.85 4 0.79 2 0.11 5 0.08 3 0.04 6 0.13 (e) Find P(X + Y = 7 and X = 2). (f) Find P(X + Y = 7 given X = 2). (g) Find P(X + Y = 10). David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 5 / 12
Example (Example 7.1) A die is rolled twice. Let X 1 and X 2 be the outcomes, and let S 2 = X 1 + X 2 be the sum of these outcomes. David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 6 / 12
Example (Example 7.1) A die is rolled twice. Let X 1 and X 2 be the outcomes, and let S 2 = X 1 + X 2 be the sum of these outcomes. Then X 1 and X 2 have the common distribution function: f (j) = 1/6 for j = 1,..., 6. Find the distribution function f S2 of S 2. David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 6 / 12
Example (Example 7.1) A die is rolled twice. Let X 1 and X 2 be the outcomes, and let S 2 = X 1 + X 2 be the sum of these outcomes. Then X 1 and X 2 have the common distribution function: f (j) = 1/6 for j = 1,..., 6. Find the distribution function f S2 of S 2. Solution: The distribution function of S 2 is then the convolution of f with itself. f S2 (j) = (f f )(j) = 6 f (k) f (j k) for j = 2,..., 12. k=1 David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 6 / 12
Figure: Density of S 10 for rolling a die 10 times. David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 7 / 12
Figure: Density of S 20 for rolling a die 20 times. David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 8 / 12
Figure: Density of S 30 for rolling a die 30 times. David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 9 / 12
Section 7.2 Sums of Continuous Random Variables Definition (Convolutions) Let X and Y be two continuous random variables with density functions f (x) and g(y), respectively. Assume that both f (x) and g(y) are defined for all real numbers. Then the convolution f g of f and g is the function given by (f g)(z) = f (z y)g(y) dx = g(z y)f (y) dx David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 10 / 12
Section 7.2 Sums of Continuous Random Variables Theorem (Theorem 7.1) Let X and Y be two independent random variables with density Let X and Y be two continuous random variables with density functions f (x) and g(y), respectively. Then the sum Z = X + Y is a random variable with density function f Z (z), where f Z (z) is the convolution of f (x) and g(y). David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 11 / 12
Section 7.2 Sums of Continuous Random Variables Example (Example 7.3) Suppose we choose independently two numbers at random from the interval [0, 1] with uniform probability density. What is the density of their sum? David Murrugarra (University of Kentucky) MA 320: Section 7.1 Spring 2017 12 / 12