#### Transcript 14 Discrete Random Variables

“Teach A Level Maths” Statistics 1 Discrete Random Variables © Christine Crisp Discrete Random Variables Suppose we roll an ordinary 6-sided die sixty times and record the number of ones, twos, etc. We might get Number on die 1 2 3 4 5 6 Frequency 12 9 11 10 7 11 If I asked you what you might expect to happen if we went on rolling the die you might say that you would expect roughly the same number of ones, twos, etc. In saying this, you would be using a perfectly reasonable model. Models in Statistics describe situations and are used to make predictions. Discrete Random Variables We could write the model out as a table: x P (X = x) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6 x gives the value of the number shown on the die. It is a variable which can be any value from 1 to 6. We let X be a description of the variable, so: “ X is the number shown on the face of the die” We label the 2nd row P (X = x) . If x = 1, for example, we get P(X = 1) which means the probability that the number shown on the die is 1. Discrete Random Variables So, we have x P(X = 1 ) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6 Discrete Random Variables So, we have x P(X = 2 ) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6 Discrete Random Variables So, we have x P(X = 3 ) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6 Discrete Random Variables So, we have x P(X = 4 ) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6 Discrete Random Variables So, we have x P(X = 5 ) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6 Discrete Random Variables So, we have x P(X = 6 ) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6 Discrete Random Variables So, we have x P(X = x ) 1 1 6 2 1 6 3 1 6 4 1 6 5 1 6 6 1 6 Discrete Random Variables So, we have x P(X = x ) 2 1 6 1 1 6 3 1 6 4 1 6 5 1 6 6 1 6 This table shows the probability distribution of X. If we add up ( sum ) the probabilities we get 1 is the Greek capital letter S and stands for Sum 6 So we can write P( X x ) 1 x 1 The sum from 1 to 6 of the probabilities of all the values of X, ( x = 1, 2, 3, 4, 5, 6 ) equals 1 Discrete Random Variables A variable where the sum of the probabilities of all its possible values is equal to 1 is called a random variable ( r.v. ). So, in our example, X is the random variable “ the number shown on the face of the die” We can usually see what values the random variable can have, so we don’t need to show them on the summation sign. So, we often write P( X x) 1 X is an example of a discrete random variable. It takes certain values only. In the example these values were the integers from 1 to 6. ( In exercises the numbers are often integers but they don’t have to be. ) Discrete Random Variables • • • • SUMMARY A statistical model uses probabilities to describe a situation and to make predictions. A probability distribution gives the probabilities for a random variable. A variable where the sum of the probabilities of all its possible values is equal to 1 is called a random variable (r.v.). If X takes only certain values in an interval, X is a discrete random variable. If X is a discrete random variable, then N.B. P( X x) 1 X describes the r.v. x gives the values of the r.v. Discrete Random Variables e.g. 1 Let X be the variable “ the number of sixes showing when 2 dice are rolled”. Show that X is a random variable and write its probability distribution in a table. Solution: We can have 0 sixes, 1 six or 2 sixes: Using a 6that ” we can 6 / forto“not We want show , so we need P ( write X x )the 1possibilities as 6 / , 6 / , 6 / , 6 of , 6getting , 6 / , 60, ,6 1 or 2 sixes. to find the probabilities 5 5 25 6 6 36 5 1 1 5 10 / / P ( X 1) P ( 6 , 6 ) P ( 6 , 6 ) 6 6 6 6 36 1 1 1 to add P ( 6be , 6 ) easier PTip: ( X It 2 ) will 6 6 36 Then, P ( X 0 ) P ( 6 / , 6 / ) the fractions if we don’t cancel Discrete Random Variables 1 10 25 P ( X 2 ) , P ( X 0) , P ( X 1) 36 36 36 36 25 10 1 1 So, P( X x) 36 36 36 36 Since P ( X x ) 1 , X is a random variable. The probability table is x P (X = x) 0 25 36 1 2 10 5 1 3618 36 Discrete Random Variables 1 10 25 P ( X 2 ) , P ( X 0) , P ( X 1) 36 36 36 36 25 10 1 1 So, P( X x) 36 36 36 36 Since P ( X x ) 1 , X is a random variable. The probability table is x P (X = x) 0 25 36 1 5 18 2 1 36 This is an example of a discrete random variable because the variable takes only some values in an interval rather than every value. Discrete Random Variables The Mean of a Discrete Random Variable We can find the mean of a discrete random variable in a similar way to that used for data. Suppose we take our first example of rolling a die. Number on die 1 2 3 4 5 6 Frequency 12 9 11 10 7 11 The mean is given by But, f1 , f2 f f xf x1 f 1 x 2 f 2 . . . x f f 1st xbe -value 1stby frequency . . . can replaced p1 , p2 . . . the probabilities of getting 1, 2, . . . So, the mean x 1 p1 x 2 p 2 . . . xp Discrete Random Variables Notation for the Mean of a Discrete Random Variable When dealing with a model, we use the letter m for the mean (the greek letter m). pronounced “mew” We write m xp or, more often, replacing p by P ( X x ) , m xP ( X x ) Instead of m, we can also write E(X). This notation comes from the idea of the mean being the Expected value of the r.v. X. Discrete Random Variables Notation for the Mean of a Discrete Random Variable When dealing with a model, we use the letter m for the mean (the greek letter m). pronounced “mew” We write m xp or, more often, replacing p by P ( X x ) , m xP ( X x ) Instead of m, we can also write E(X). This notation comes from the idea of the mean being the Expected value of the r.v. X. ( Think of this as being what we expect to get on average ). Discrete Random Variables e.g. 1. A random variable X has the probability distribution x 1 5 10 1 1 p P (X = x) 4 2 Find (a) the value of p and (b) the mean of X. Solution: (a) Since X is a discrete r.v., p1 p 14 (b) mean, m xP ( X x ) 1 14 5 12 10 14 1 4 P( X x) 1 1 2 21 4 Tip: Always check that your value of the mean lies within the range of the given values of x. Here, 21 4 or 5·25, does lie between 1 and 10. Discrete Random Variables The probabilities in a probability distribution can sometimes be given by a formula. The formula is called a probability density function ( p.d.f. ). e.g. 1. Write out a probability distribution table for the r.v. X where x P( X x) 6 Solution: x P (X = x) 1 1 6 x 1, 2, 3 for 2 2 6 1 3 3 3 6 1 2 Discrete Random Variables The probabilities in a probability distribution can sometimes be given by a formula. The formula is called a probability density function ( p.d.f. ). e.g. 1. Write out a probability distribution table for the r.v. X where x P( X x) 6 Solution: x P (X = x) 1 1 6 for 2 1 3 x 1, 2, 3 3 1 2 These probabilities can be shown on a diagram. Discrete Random Variables x P (X = x) 1 1 6 2 1 3 3 1 2 P( X x) This is called a stick diagram. 1 2 1 3 1 6 x 1 2 3 Discrete Random Variables e.g. 2. Find the value of the constant k for the random variable X with p.d.f. given by P ( X x ) kx for x 1, 2, 3, 4 Solution: Since X is a discrete random variable, So, P( X x) 1 1k 2 k 3 k 4 k 1 10k 1 k 01 Discrete Random Variables SUMMARY • The mean, m , of a discrete random variable is given by m xP ( X x ) • • The mean is also referred to as the expectation or expected value of the r.v. m can be written as E(X) • The probabilities can be given by a formula called the probability density function ( p.d.f. ) • An unknown constant in the p.d.f. can be found by using P( X x) 1 Discrete Random Variables Exercise 1. The tables show the probability distributions of 2 random variables. For each, find (i) the value of p (ii) the mean value. (b) (a) x P (X = x) 1 1 6 2 1 3 x 3 0 P (X = x) 0 3 p 1 2 p 0 6 2. Write out the probability distribution for the random variable, X, where the probability distribution function is x P( X x) for x 1, 2, 3, 4 10 Discrete Random Variables Solution: 1(a) X is a random variable: x P (X = x) mean, (b) m 1 1 6 2 1 3 3 p P( X x) 1 1 1 1 p1 p 6 3 2 1 1 1 xP ( X x ) m 1 2 3 6 3 2 14 7 6 3 x 0 P (X = x) 0 3 1 2 p 0 6 03 p 06 1 p 01 mean, m 0 0 3 1 0 1 2 0 6 1 3 Discrete Random Variables 2. Write out the probability distribution for the random variable, X, where the probability distribution function is Solution: x P( X x) 10 for x 1, 2, 3, 4 x 1 2 3 P(X = x ) 1 10 21 10 5 3 10 4 42 10 5 Discrete Random Variables 2. Write out the probability distribution for the random variable, X, where the probability distribution function is Solution: x P( X x) 10 for x 1, 2, 3, 4 x 1 2 3 4 P(X = x ) 1 10 1 5 3 10 2 5 Discrete Random Variables Exercise 3. Find the exact value of the constant k for the random variable X with p.d.f. given by k P( X x) x Solution: for x 1, 2, 3 Since X is a discrete random variable, P( X x) 1 So, k k k 1 1 2 3 6k 3k 2k 1 6 11k 1 6 6 k 11 Discrete Random Variables Variance of a Discrete Random Variable The variance of a discrete random variable is found in a similar way to the one we used for the mean. For a frequency distribution, the formula is 2 x f 2 2 variance s x f x f1 x f 2 . . . x2 f 2 Replacing f1 2 by p1 etc. gives f s x 1 p1 x 2 p 2 . . . x 2 2 2 2 Discrete Random Variables s 2 x 1 p1 x 2 p 2 . . . x 2 2 2 But we must replace x by m and we replace s by the letter ( which is the Greek lowercase s, pronounced sigma ). So, 2 x 2 p1 x 2 p 2 . . . m 2 2 x 2 P( X x) m 2 ( Notice that this expression contains the Greek capital S, S, and the lowercase s, . ) The variance of X is also written as Var(X). Discrete Random Variables e.g. 1 Find the variance of X for the following: x 1 2 3 4 P(X = x ) 1 10 2 10 3 10 4 10 Solution: 2 x 2 P( X x) m 2 We first need to find the mean, m : m xP ( X x ) 30 1 2 3 4 3 m 1 2 3 4 10 10 10 10 10 2 x 2 P( X x) m 2 1 2 3 4 2 2 2 1 2 3 4 ( 3) 2 10 10 10 10 Tip: With a1 bit8of 27 practice 64 you’ll find 100 you can simplify 9 quicker 1 without calculator. 9 the fractions a It’s and 10 10 10 10 10 more accurate. Try these before you see my answer. 2 2 Discrete Random Variables • SUMMARY The mean of a discrete random variable is given by E( X ) m xP ( X x ) • The variance, , of a discrete random variable is given by 2 Var ( X ) 2 x 2 P ( X x ) m 2 N.B. For frequency distributions use x and s 2 for the mean and variance ( the “English” alphabet ). For probability distributions ( models ) use m and 2 ( the Greek alphabet ). Discrete Random Variables Exercise 1. Find the variance of X for each of the following: (b) (a) x P (X = x) 1 1 6 Solution: (a) 2 2 2 6 x 3 3 6 0 1 2 P (X = x) 0 3 0 1 0 6 m xP ( X x ) 7 1 2 3 14 7 1 2 3 6 6 6 63 3 2 1 2 3 7 x 2 P ( X x ) m 2 12 2 2 3 2 6 6 6 3 1 8 27 49 49 5 6 6 6 6 9 9 9 Discrete Random Variables Exercise (b) x 0 1 2 P (X = x) 0 3 0 1 0 6 Solution: m xP ( X x ) 1 0 1 2 0 6 1 3 2 x 2 P( X x) m 2 12 0 1 2 2 0 6 1 3 2 2 5 1 69 0 81 Discrete Random Variables 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. Discrete Random Variables • • • • SUMMARY A statistical model uses probabilities to describe a situation and to make predictions. A probability distribution gives the probabilities for a model. A variable where the sum of the probabilities of all its possible values is equal to 1 is called a random variable (r.v.). If X takes only certain values in an interval, X is a discrete r.v. If X is a discrete random variable ( r.v. ), then N.B. P( X x) 1 X describes the r.v. x gives the values of the r.v. Discrete Random Variables The Mean of a Discrete Random Variable We can find the mean of a discrete random variable in a similar way to that for data. Suppose we take our first example of rolling a die. Number on die 1 2 3 4 5 6 Frequency 12 9 11 10 7 11 The mean is given by But, f1 , f2 f f xf x1 f 1 x 2 f 2 . . . x f f . . . can be replaced by p1 , p2 . . . the probabilities of getting 1, 2, . . . So, the mean x 1 p1 x 2 p 2 . . . px Discrete Random Variables Notation for the Mean of a discrete Random Variable When dealing with a discrete random variable, we use the letter m ( pronounced mew ) for the mean (the greek letter m). We write m xp or, more often, replacing p by P ( X x ) , m xP ( X x ) Instead of m, we can also write E(X). The notation comes from the idea of the mean being the Expected value of the r.v. X. ( Think of this as being what we expect to get on average ). Discrete Random Variables e.g. 1 Let X be the variable “ the number of sixes showing when 2 dice are rolled”. Show that X is a random variable and write its probability distribution in a table. Solution: We can have 0 sixes, 1 six or 2 sixes: Using 6 / for “ not a 6 “ we can write the possibilities as 6 / , 6 / , 6 / , 6 , 6, 6 / , 6, 6 5 5 25 6 6 36 5 1 1 5 10 / / P ( X 1) P ( 6 , 6 ) P ( 6 , 6 ) 6 6 6 6 36 1 1 1 P ( X 2 ) P ( 6, 6 ) 6 6 36 Then, P ( X 0 ) P ( 6 / , 6 / ) Discrete Random Variables 1 10 25 P ( X 2 ) , P ( X 0) , P ( X 1) 36 36 36 36 25 10 1 1 So, P( X x) 36 36 36 36 The probability table is x P (X = x) 0 25 36 1 5 18 2 1 36 Discrete Random Variables SUMMARY • The mean, m , of a discrete random variable is given by m xP ( X x ) • • • The mean is also referred to as the expectation or expected value of the r.v. m can be written as E(X) The probabilities can be given by a formula called the probability density function ( p.d.f. ) Discrete Random Variables Variance of a Discrete Random Variable For a frequency distribution, the formula is variance s Replacing f1 f 2 f fx 2 x2 f1 x 2 f 2 x 2 . . . x2 f by p1 etc. gives s p1 x 1 p 2 x 2 . . . x 2 2 2 2 But for a random variable, we must replace x by m and we replace s by the letter (the greek s, pronounced sigma ). So, 2 p1 x 2 p 2 x 2 . . . m 2 2 x 2 P( X x) m 2 The variance of X is also written as Var(X). Discrete Random Variables e.g. 1 Find the variance of X for the following: x 1 2 3 4 P(X = x ) 1 10 2 10 3 10 4 10 2 x 2 P( X x) m 2 Solution: xP ( X x ) We first need to find the mean, m : m 30 1 2 3 4 3 m 1 2 3 4 10 10 10 10 10 2 x 2 P( X x) m 2 1 2 3 4 2 2 2 1 2 3 4 ( 3) 2 10 10 10 10 1 8 27 64 100 91 9 10 10 10 10 10 2 2 Discrete Random Variables SUMMARY • The mean, m , of a discrete random variable is given by E( X ) m xP ( X x ) • The variance, , of a discrete random variable is given by 2 Var ( X ) 2 x 2 P ( X x ) m 2 N.B. For frequency distributions use x and s 2 for the mean and variance ( the “English” alphabet ). For probability distributions ( models ) use m and 2 ( the Greek alphabet ).