Discrete Random Variables

1
Discrete Random Variables

• A random variable (r.v.) assigns a numerical
  value to the outcomes in the sample space of a
  random phenomenon.
• A discrete r.v X has a finite number of possible
  values. The probability distribution of X lists
  the values xi and their probabilities pi. Every
  pi is a number between 0 and 1. The sum of the
  pis must equal 1.
• Examples
• 1. Consider the experiment of tossing a coin.
  Define a random variable as follows X 1 if a
  H comes up
• 0 if a
  T comes up.
• This is an example of a Bernoulli r.v. The
  Probability function of X is given in the
  following table

x P(X x)
1 0 p 1- p
2

• Let X be a r.v counting the number of girls in a
  family with 3 children.
• The probability function of X is given
  in the following table.
• Toss a coin 4 times. Let X be the number of Hs.
  Find the probability function of X. Draw a
  probability histogram.
• Toss a coin until the 1st H. Let X be the number
  of Ts before the 1st H. Find the probability
  function of X.

x P(X x)
0 1 2 3 (0.5)3 0.125 3(0.5)3 0.375 3(0.5)3 0.375 (0.5)3 0.125
3
Continuous random variables

• A continuous r. v. X takes all values in an
  interval of numbers.
• The probability distribution of X is described by
  a density curve.
• The total area under a density curve is 1.
• The probability of any event is the area under
  the density curve and above the value of X that
  make up the event.
• Example
• The density function of a continuous r. v. X
  is given in the graph below.
• Find i) P(X lt 7)
• ii) P(6 lt X lt 8)
• iii) P(X 7)
• iv) P(5.5 lt X lt 7 or 8 lt X lt 9)

4
Normal distributions

• The density curves that are most familiar to
  us are the normal curves.

5
Mean (expected value) of a discrete r. v.

• The mean of a r. v. X is denoted by µx and can
  be found using the following formula
• Examples
• 1. The mean of the Bernoulli r.v defined in
  example 1 above is
• µx 0(1-p) 1p p
• 2. The mean number of girls in a family with 3
  children is 1.5.
• Exercise Find the mean of X in example 3 above.
• Exercise Read Example 4.20 on p291 in IPS.

6
Example Discrete Uniform r.v

• Roll a six-sided die. Define a r. v. X to be the
  number shown on the die. That is, X 1 if die
  lands on 1,
• X 2 if die lands
  on 2, etc.
• The probability distribution of X is given
  in the table below
• The mean of X is
• µX 1(1/6) 2(1/6) 3(1/6) 4(1/6)
  5(1/6) 6(1/6) 21/6 3.5 .

x P(X x)
1 2 3 4 5 6 1/6 1/6 1/6 1/6 1/6 1/6
7
Law of large numbers

• If independent observations are drawn from a
  population with a finite mean ?, the population
  mean ? can be estimated with a specified degree
  of accuracy by the sample mean , using
  sufficiently large sample.

8
Rules for Mean of r.v

• For any two r.vs X and Y and constants a and b,
• 1. µ x a µx a .
• 2. µ bx bµx .
• 3. µ bx a bµx a .
• 4. µ X Y µX µY.
• Example
• The price X of Nike sports shoes is a random
  variable with mean µx 200.
• Before the holidays Nike company had a
  promotion Pay 10 less for each
• item and get 20 discount from the original
  price.
• What is the mean price during the promotion.
• Suppose in addition that during the promotion the
  mean price for Nike socks is µY 20. What is
  the expected value of your expenses if you are to
  buy one pair of shoes and one pair of socks ?

9
The variance of a r. v.

• The variance of a r. v. is an average of the
  squared deviations from the mean, (X µx)2 .
• The Variance of a discrete r. v. is
• The standard deviation sX of a r. v. is the
  square root of its variance.
• Examples
• The variance of a Bernoulli r.v is
• s2x p p2 p(1- p)
• 2. The variance of the Uniform example above
  is
• s2x (91/6) (3.5)2 2.9167

10
Rules for variances

• If X is a r. v. and a and b are constants, then
• 1.
• 
  
  
• 2.
• 3.
• 4. If X and Y are independent r. vs then,

11

• Two random variables X and Y are independent if
  knowing that any event involving X alone did or
  did not occur tells us nothing about the
  occurrence of any event involving Y alone.
• Example
• Consider again the Nike example above. If
  the stdev. of X is sx 10 and the stdev. of Y
  is sY 8.
• What is the stdev. of the shoes price during the
  promotion? (8).
• What is the stdev. of your expenses if you were
  to buy one pair of shoes and one pair of socks ?
  (12.806).
• Example 4.34 on page 283 in IPS.