ECMT1010 Lecture Notes - Lecture 12: Standard Deviation, Binomial Coefficient, Random Variable
13 Aug 2018
School
Department
Course
Professor
ECMT1010
Lecture 12 Notes
Lecturer: Tim Fisher
Random Variable:
A random variable denotes a numeric quantity that changes from trial to trial in a random
process.
Examples:
X = number of home team wins in NBA
Y = sum of two dice rolls
T = time to run 1500m
W = weight of a rat
Discrete v Continuous random variables:
A random variable is discrete if it has a finite set of possible values.
X = number of home team wins in NBA = {0, 1, 2, …}
Y = sum of two dice rolls {2, 3, …, 12}
A random variable is continuous if it has values within some interval.
T = time to run 1500m
W = weight of a rat
Probability Function – Discrete Random Variables:
For a discrete random variable, a probability function assigns a probability, between 0 and
1, to every value of a discrete random variable. The sum of all these probabilities must be
one, i.e.
Example: Suppose that we roll two six-sided dice and let a random variable X measure the
sum of the two rolls. The probability function for X is shown in the table below.
Use the probability function to find:
a) P(X = 7 or X = 11)
b) P(X > 8)
a) The events X = 7 and X = 11 are disjoint so we find the probability that one or the
other occurs by adding the individual probabilities:
b) To find the probability that the sum is greater than 8, we add the individual
probabilities from the probability function for the values of X that satisfy this
condition:
Mean of a Random Variable:
The mean (expected value) for a discrete random variable X with probability function p(x),
the mean is:
Example:
A $1 lottery game has prizes given by
X
$55
$5
$1
$0
p(x)
0.003
0.043
0.213
0.741
a) Find the mean prize for a single ticket.
b) Is it worth the $1 cost to play?
On average, you would lose about 41 cents for every $1 ticket.
Variance and Standard Deviation:
For a random discrete variable with probability function p(x) and mean, , the variance is:
The standard deviation is given by:
Document Summary
A random variable denotes a numeric quantity that changes from trial to trial in a random process. X = number of home team wins in nba. A random variable is discrete if it has a finite set of possible values. X = (cid:374)u(cid:373)(cid:271)er of ho(cid:373)e tea(cid:373) wi(cid:374)s i(cid:374) nba = {(cid:1004), (cid:1005), (cid:1006), } Y = su(cid:373) of two di(cid:272)e rolls {(cid:1006), (cid:1007), , (cid:1005)(cid:1006)} A random variable is continuous if it has values within some interval. For a discrete random variable, a probability function assigns a probability, between 0 and. 1, to every value of a discrete random variable. The sum of all these probabilities must be one, i. e. (cid:4666)(cid:4667)= Example: suppose that we roll two six-sided dice and let a random variable x measure the sum of the two rolls. The probability function for x is shown in the table below. The mean (expected value) for a discrete random variable x with probability function p(x), the mean is:
