Teach A Level Statistics Maths 1 Cumulative Distribution

“Teach A Level Statistics Maths” 1 Cumulative Distribution Function © Christine Crisp

Cumulative Distribution Function Statistics 1 Edexcel "Certain images and/or photos on this presentation are the copyrighted property of Jupiter. Images and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from Jupiter. Images"

Cumulative Distribution Function When we met frequency distributions, we sometimes found the cumulative frequencies. e. g. x f Cu. f. 1 5 5 2 8 13 3 10 23 4 12 35 5 15 50 In a similar way, with a probability distribution, we can find cumulative probabilities. e. g. 1 x 1 0· 1 2 0· 3 3 0· 4 0· 7 4 0· 2 0· 9 5 0· 1 1 We call the distribution of cumulative probabilities the Cumulative Distribution Function.

Cumulative Distribution Function e. g. 1 For a discrete random variable X the p. d. f. is given by for where k is a constant. Write out a table showing the probability distribution and the cumulative distribution function. Find a formula for F(x), the cumulative distribution function. Solution: The sum of the probabilities is 1, so x 1 0· 25 2 0· 25 3 0· 25 4 0· 25

Cumulative Distribution Function e. g. 1 For a discrete random variable X the p. d. f. is given by for where k is a constant. Write out a table showing the probability distribution and the cumulative distribution function. Find a formula for F(x), the cumulative distribution function. Solution: The sum of the probabilities is 1, so x 1 0· 25 2 0· 25 0· 5 3 0· 25 0· 75 4 0· 25 1

Cumulative Distribution Function e. g. 2 For a discrete r. v. X the cumulative distribution function F(x) is given in the table. x F(x) 0 1 2 3 4 5 0· 05 0· 1 0· 35 0· 65 1 Find (a) P(X = 4) Solution: (a) (b) P(X > 3)

Cumulative Distribution Function e. g. 2 For a discrete r. v. X the cumulative distribution function F(x) is given in the table. x 0 1 2 3 4 5 F(x) 0· 05 0· 1 0· 35 0· 65 1 Find (a) P(X = 4) (b) P(X > 3) Solution: (a) -

Cumulative Distribution Function e. g. 2 For a discrete r. v. X the cumulative distribution function F(x) is given in the table. x F(x) 0 1 2 3 4 5 0· 05 0· 1 0· 35 0· 65 1 Find (a) P(X = 4) (b) P(X > 3) Solution: (a) (b)

Cumulative Distribution Function e. g. 2 For a discrete r. v. X the cumulative distribution function F(x) is given in the table. x 0 1 2 3 4 5 F(x) 0· 05 0· 1 0· 35 0· 65 1 Find (a) P(X = 4) (b) P(X > 3) Solution: (a) (b)

Cumulative Distribution Function e. g. 3. A discrete random variable X has a p. d. f. given by where k is a constant. (a) Find the value of k. (b) Write out a table showing the probability distribution and the cumulative distribution function. (c) Find a formula for F(x), the cumulative distribution function. Solution: (a) Since X is a r. v. ,

Cumulative Distribution Function e. g. 3. A discrete random variable X has a p. d. f. given by (b) Write out a table showing the probability distribution and the cumulative distribution function. Solution: x 1 2 3 P(X = x) k 3 k 5 k

Cumulative Distribution Function e. g. 3. A discrete random variable X has a p. d. f. given by (b) Write out a table showing the probability distribution and the cumulative distribution function. Solution: x 1 2 3 P(X = x) Now we can add the cumulative probabilities.

Cumulative Distribution Function e. g. 3. A discrete random variable X has a p. d. f. given by (b) Write out a table showing the probability distribution and the cumulative distribution function. Solution: x 1 2 3 P(X = x) F(x) (c) The cumulative distribution function, F(X) is given by

Cumulative Distribution Function SUMMARY The Cumulative Distribution Function is given by F(x) where e. g. If a p. d. f. is defined for x = 1, 2, 3, . . . n then,

Cumulative Distribution Function Exercise 1. A discrete random variable X has a p. d. f. given by where k is a constant. (a) Find the value of k. (b) Write out a table showing the probability distribution and the cumulative distribution function.

Cumulative Distribution Function 1. A discrete random variable X has a p. d. f. given by where k is a constant. (a) Find the value of k. (b) Write out a table showing the probability distribution and the cumulative distribution function. Solution: (a) Since X is a random variable,

Cumulative Distribution Function 1. A discrete random variable X has a p. d. f. given by where k is a constant. (a) Find the value of k. (b) Write out a table showing the probability distribution and the cumulative distribution function. Solution: (b) x 0 1 2 3 4 P(X = x) k 2 k 3 k 4 k 5 k F(X)

Cumulative Distribution Function 1. A discrete random variable X has a p. d. f. given by where k is a constant. (a) Find the value of k. (b) Write out a table showing the probability distribution and the cumulative distribution function. Solution: (b) x P(X = x) F(X) 0 1 2 3 4

Cumulative Distribution Function The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

Cumulative Distribution Function SUMMARY The Cumulative Distribution Function is given by F(x) where e. g. If a p. d. f. is defined for x = 1, 2, 3, . . . n then,

Cumulative Distribution Function e. g. 1 For a discrete random variable X the p. d. f. is given by for where k is a constant. Write out a table showing the probability distribution and the cumulative distribution function. Find a formula for F(x), the cumulative distribution function. Solution: The sum of the probabilities is 1, so x 1 0· 25 2 0· 25 0· 5 3 0· 25 0· 75 4 0· 25 1

Cumulative Distribution Function e. g. 2 For a discrete r. v. X the cumulative distribution function F(x) is given in the table. x 0 1 2 3 4 5 F(x) 0· 05 0· 1 0· 35 0· 65 1 Find (a) P(X = 4) (b) P(X > 3) Solution: (a) (b)