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Discrete random variable
Compiled by: Rashid Ali Qureshi
Curriculum Content of CIE:
Candidate should be able to
 Construct a probability distribution table relating to a given situation involving a discrete
random variable X, and calculate E(X) and Var(X);
 Use formulae for probabilities for the binomial distribution, and recognize practical situations
 Formulae for the expectation and variance of the binomial distribution.
Topics:
Discrete Random variables: A random variable is a function or rule that assigns a number to each
outcome of an experiment. Basically it is just a symbol that represents the outcome of an
experiment.
Examples:
i.
A coin is tossed three times and X is the number of tails in three tosses then X is a discrete
variable.
ii.
Y is the number of sixes when a dice is thrown twice, so Y is a discrete random variable.
Notation:
An upper-case letter will represent the name of the random variable, usually X. Its lower-case
counterpart, x, will represent the value of the random variable. The probability that the random
variable 𝑿 will equal x is:
𝑃(𝑿 = 𝑥) or more simply 𝑃(𝑥)
X = number of heads in 10 flips of coin
P(X = 5) = P (5) = probability of 5 heads (x) in 10 flips.
Probabilities,𝑃(𝑋 = 𝑥), associated with Discrete random variables have the following properties.
i.
0 ≤ 𝑃(𝑋 = 𝑥) ≤ 0 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥
∑ 𝑃(𝑋 = 𝑥) = 1 or
∑ 𝑝𝑖 = 1
ii.
Probability distribution function (p.d.f): The arrangement of values of discrete random variable with
their probabilities in a table is called the probability distribution. To find the probabilities of two or
more than two values then their probabilities are added.
Examples
A coin is tossed twice and X is the number of heads in two tossed then probability
distribution is as
0
1
2
𝑋=𝑥
0.25
0.5
0.25
𝑃(𝑋 = 𝑥)
Q.1
Two tetrahedral dice, each with a faces labeled 1, 2, 3 and 4 are thrown and the score is
noted, where the score is sum of two numbers on which the dice land. X is the score when two
dice are thrown. Find the probability distribution function of X.
Q.2
The p.d.f of a discrete random variable y is
0
1
2
3
4
𝑌=𝑦
c
4c
9c
16c
𝑃(𝑌 = 𝑦) 0
Find the value of c and calculate 𝑃(𝑌 = 3), 𝑃(0 ≤ 𝑌 ≤ 3) and 𝑃(𝑋 > 1).
Expected value or Expectation or mean of discrete random variable :Expected value of discrete
variable X is denoted by 𝐸(𝑋) and given by
𝝁 = 𝐸(𝑋) = ∑ 𝑝𝑖 𝑥𝑖 where 𝑝𝑖 is the probability of 𝑥𝑖
Expected value gives us the mean of discrete random variable.
Compiled By : Sir Rashid Qureshi
www.levels.org.pk
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Variance of discrete random variable: The variance of discrete random variable is denoted by
Var (X) and is given by
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𝑉𝐴𝑅 (𝑋) = 𝑉(𝑋) = 𝐸(𝑋 2 ) − (𝐸(𝑋))
Where 𝐸(𝑋) is the expected value of discrete random variable and E(X 2 ) = ∑ pi (xi )2
Q.3
The discrete random variable X has p.d.f as
𝑋=𝑥
𝑃(𝑋 = 𝑥)
1
0.2
2
0.3
3
0.5
Find the mean and variance of X.
Q.4
Two fair dice are thrown. Let the random variable X be the smaller of two scores if the
scores are different, or the score on one of the die if the scores are same.
i.
Copy and complete the following table to show the probability distribution of X
1
2
3
4
5
6
𝑋=𝑥
𝑃(𝑋 = 𝑥)
ii.
ii.
Find 𝐸(𝑋).
Q.5
A fair cubical dice with faces 1, 1, 1, 2,,3,4 is thrown and score is noted. The area 𝐴 of a
square of side equal to score is calculated, so, for example, when the score on die is 3, the value
of 𝐴 is 9.
 Draw up a table to show the probability distribution of 𝐴.
 Find the mean and variance of 𝐴.
Q.6
A box contains 10 pens of which 3 are new. A random sample of two pens is taken.
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a. Show that the probability of getting exactly one new pen in the sample is .
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b. Construct a probability distribution table for the number of new pens in the sample.
c. Calculate the expected number of new pens in the sample.
Q.7
A fair dice has four faces. One face is coloured pink one is coloured orange, one is
coloured green and one is coloured black. Five such dice are thrown and the number that fall on
green face is counted. The random variable X is the number of dice that fall on a green face.
i. Show that the probability of 4 dice landing on green face is 0.0146, correct to 4 decimal
places.
ii. Draw up a table for the probability distribution of X, giving your answers correct to 4
decimal places.
Q.8
A fair dice has one face numbered 1, one face numbered 3, two faces numbered 5 and
two faces numbered 6. The dice is thrown twice. Let X is the sum of two scores.
i. Draw up a table showing the probability distribution of 𝑋.
ii. Calculate𝐸(𝑋).
iii. Find the probability that X is greater than 𝐸(𝑋).
iv. Calculate 𝑉𝑎𝑟 ( 𝑋).
Q.9
The discrete random variable X has the following distribution.
0
1
2
3
4
𝑋=𝑥
0.05
0.09
𝑃(𝑋 = 𝑥) 0.26
𝑞
3𝑞
i. Find the value of 𝑞.
Find the 𝐸(𝑋) and 𝑉𝑎𝑟 ( 𝑋).
Compiled By : Sir Rashid Qureshi
www.levels.org.pk
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