A discrete uniform distribution is described where each outcome has an equal probability of 1/n. Examples of calculating the probability mass function, expected value, and variance are provided. Tasks involve computing probabilities, means, and variances for discrete uniform distributions. The document then summarizes the binomial distribution and provides examples of calculating probabilities using the binomial probability formula. Tasks involve computing probabilities for binomial distributions. Next, the Poisson distribution is described and examples are given for calculating probabilities. Tasks involve finding probabilities for Poisson distributions. Approximating binomial distributions with Poisson distributions when n is large and np is small is also covered. The document concludes with defining the continuous uniform distribution and providing examples of computing probabilities, expected values, and variances for continuous uniform random

1. SPECIAL PROBABILITY
DISTRIBUTION

2. Discrete Uniform Distribution
A discrete random variable X is said to have uniform distribution, if its
probability function has constant value and defined as
𝑓 𝑥 = 𝑝, 𝑓𝑜𝑟 𝑥 = 𝑥1, 𝑥2, … . 𝑥𝑛
Example: A fair dice is thrown, the number obtained is a random variable X
giving a uniform distribution.
X 1 2 3 4 5 6
P(X=x) 1/6 1/6 1/6 1/6 1/6 1/6

3. If X is a discrete random variable with a uniform distribution, then the
a) Probability mass function (pmf): 𝑃 𝑋 = 𝑥 =
1
𝑛
; 𝐴 ≤ 𝑥 ≤ 𝐵
0 ; 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
b) Probability : 𝑃 𝑋 = 𝑥𝑖 =
1
𝑛
for i = 1,2,3,…,n
c) Expected Value: 𝐸 𝑋 = 𝑥 𝑃 𝑋 = 𝑥 or 𝐸 𝑋 =
𝐴+𝐵
2
d) Variance: 𝑉𝑎𝑟 𝑋 = 𝐸 𝑋2 − 𝐸(𝑋) 2 or 𝑉𝑎𝑟 𝑋 =
𝑛2−1
12
where 𝑛 = 𝐵 − 𝐴 + 1

4. Example 1
An unbiased spinner, numbered 1, 2, 3, 4 is spun. X (the random variable) is
the number obtained. Find:
a) The probability distribution function.
b) Mean of the function
c) Variance of the function

5. Example 2
Roll a dice. Find the
a) probability that an odd number appear when rolling the dice.
b) Probability that the number appear when rolling the dice is less than 4.
c) Expected value and variance.

6. Task 1
For the following discrete uniform distribution function:
𝑓 𝑥 =
1
𝑛
, 𝑓𝑜𝑟 𝑥 = −3, −2, −1, 0,1,2,3
a) Show that n = 7
b) Calculate the mean and variance.

7. Task 2
A discrete random variable x has the following probability.
Find:
a) 𝑃 2 < 𝑥 < 10
b) 𝑃 𝑥 ≥ 5
c) Mean and standard deviation.
x 2 4 6 8 10
P(X = x) 0.2 0.2 0.2 0.2 0.2

8. Task 3
For the following discrete uniform distribution function
𝑓 𝑥 =
1
𝑛
, 𝑓𝑜𝑟 𝑥 = 1,2,3, … , 8
Find:
a) 𝑃 2 ≤ 𝑋 < 4
b) 𝑃(𝑋 ≥ 5)
c) Standard deviation

9. BINOMIAL DISTRIBUTION

10. Introduction

16. Example 2:
Use the binomial probability formula to find the probability of 2 successes in 9 trials.
Given the probability of success is 0.35
Example 3:
Given that 𝑋~𝐵(8,0.2). Find
a) 𝑃 𝑋 = 6
b) 𝑃(𝑋 ≤ 1)
Example 4:
Given that 𝑋~𝐵(12,0.4). Find
a) 𝑃 𝑋 = 3
b) 𝑃(𝑋 > 1)

17. Task 1:
Let X be distributed to binomial distribution with p = 0.35 and n = 8. find the probability that
less than three are success.
Task 2:
Given that the probability of success, p = 0.25 and the number of independent trial, n = 6. find
the probability that
a) Exactly three b) at least three c) at most two
Task 3:
The probability that a baby is born a boy is 0.51. a mid-wife delivers 10 babies. Find the
probability that:
a) Exactly 4 are male b) at least 8 are male

19. Example 1
If X ~ B(16, 0.25), find E(X) and Var (X).
Example 2
Find the mean and the variance for the Binomial distribution if p = 0.45 and n = 10.
Example 3
If X~ B(n, p), E(X) = 8 and Var (X) = 4.8, calculate P(X = 5).

20. TASK
1. A biased dice is rolled. The probability that a ‘5’ occurs is
1
4
. If the dice is rolled
200 times, find
a) The mean
b) The standard deviation, that the number ‘5’ occurs.
2. A random variable X has a Binomial distribution with parameters n and p. The
mean value is 5.28 and the variance is 2.96. Find the value of n and p.

21. Using the Table of Binomial Probabilities.
Guidelines when p ≤ 𝟓:
Let us check together how the table can be used!

22. Example
Given that n = 10 and p = 0.05. By using statistical table, find:
a) 𝑃 𝑋 ≥ 2
b) 𝑃 𝑋 > 2
c) 𝑃 𝑋 < 2
d)𝑃 𝑋 = 2
e) 𝑃(2 < 𝑋 < 5)

23. TASK
Please complete the task in your module page 110 – 112
Task 4.5, Task 4.6, Task 4.7, Task 4.8 and Task 4.9

24. CALCULATING PROBABILITIES WHEN p > 0.5
Sometimes the binomial event has the probability of success of more than 0.5.
So, how can this be solved??
Guideline when p > 0.5

25. Example 1
If X~B(12, 0.8), find:
a) P(X = 6) c) 𝑃(𝑋 ≤ 4)
b) P(X > 10) d) 𝑃(2 < 𝑋 < 6)
Example 2
A traffic control engineer reports that 25% of the vehicles passing through
a checkpoint are from State JK. What is the probability that more than 2
of the next 9 vehicles are from outside State JK?

26. Do it yourself
Task 1
If X ~ B(10, 0.6), find using the binomial table.
a) 𝑃 𝑋 ≥ 3 = 0.9887 d) 𝑃 𝑋 > 5 = 0.6331
b) 𝑃 𝑋 ≤ 5 = 0.3669 e) 𝑃(1 < 𝑋 ≤ 4) = 0.1645
c) 𝑃(𝑋 = 4) = 0.1114 f) 𝑃(2 ≤ 𝑋 < 8) = 0.8310

27. Do it yourself
Task 2
In a survey it was found out that 60% of the students in a college have hand
phones. If 20 students are picked at random, find the probability that the number of
students that have hand phones is
a) More than 10
b) At least 12
c) Between 7 and 12
d) Less than 14 Ans: a) 0.7553, b) 0.5956 c) 0.3834 d) 0.7500
Please complete task 4.10, task 4.11 and task 4.12 in your
module page 113 – 114.

28. POISSON DISTRIBUTION

29. • POISSON DISTRIBUTION

33. Example
1. Suppose 𝑋~𝑃0(0.85). Find
a) P(X = 3)
b) P(X > 2)
2. Suppose that, on average, 5 cars enter a parking lot per minute. What is the probability that
in a given minute, 7 cars will enter?
3. On average a call centre receives 1.75 phone calls per minute.
a) assuming a Poisson distribution, find the probability that the number of phone calls
received in a randomly chosen minute is:
i) exactly 4 ii) not more than 2
b) Find the probability that 6 phone calls are received in a 4 minute period.

34. TASK
If 𝑋~𝑃0(3), find:
a) 𝑃(𝑋 = 2) e) 𝑃(2 < 𝑋 ≤ 5)
b) 𝑃(𝑋 ≥ 3) f) the mean
c) 𝑃 𝑋 ≤ 3 g) the variance
d) 𝑃(𝑋 < 2) h) the standard deviation
Complete task 4.13 and 4.14 in your module page 116.

35. Using the Table of Poisson Probabilities
Suppose X has a Poisson distribution with mean of 9.9. Determine the following probabilities:
a) 𝑃 𝑋 ≤ 11
b) 𝑃 𝑋 > 6
c) 𝑃(𝑋 < 18)

36. TASK
If 𝑋~𝑃0(3), find:
a) 𝑃(𝑋 = 2)
b) 𝑃(𝑋 ≥ 3)
c) 𝑃 𝑋 ≤ 3
d) 𝑃(𝑋 < 2)
e) 𝑃(2 < 𝑋 ≤ 5)
Complete task 4.15 until 4.20 in your module page 117 - 118.

37. Poisson Approximation of Binomial Probabilities
The Poisson distribution can be used to approximate the Binomial distribution when the
number of trial, 𝑛 ≥ 30 and 𝑛𝑝 < 5.
Definition
If X is a binomial random variable with (n, p), then 𝑛 ≥ 30 and 𝑛𝑝 < 5, then 𝑋~𝑃0(𝑛𝑝).
Example:
For a large shipment of papayas, 2% of the papayas are rotten. If the a sample of 50
papayas is randomly selected, find the probability that 5 or more papayas will be rotten.

38. Task 1
A drug manufacturer has found that 2% of patients taking a particular drug
will experience a particular side-effect. A hospital consultant prescribes the
drug to 150 of her patients. Using a suitable approximation, calculate the
probability that:
a) None of her patients suffer from the side-effects.
b) Not more than 5 suffer from the side-effects.
Ans: a) 0.0498 b) 0.9161

39. Please complete task 4.21 – 4.24 in your module page 119 – 120.
Task 2
There are 500 eggs in a box. At an average, 0.06% are broken before it
arrives at a supermarket. Find the probability in a box containing 500
eggs,
a) Exactly 3 eggs are broken
b) Less than 2 eggs are broken
c) At least 3 eggs are broken ans: a) 0.0033 b) 0.9631 c) 0.0036

42. Continuous Uniform Distribution

43. Example:
X is a uniformly distributed continuous random variable, such that 𝑋~𝑅(3,8). Find:
a) The probability density function of X.
b) 𝑃 3.5 ≤ 𝑋 ≤ 7 and 𝑃(𝑋 > 5)

45. Example
If 𝑋~𝑅(2,5) write down the probability density function of X. Then find:
a) E(X)
b) Var (X)

46. Task
1. The continuous random variable X has a uniform distribution such that 𝑋~𝑅(2.4, 4.8).
Find:
a. The probability density function of X
b. E(X) and Var(X)
c. 𝑃 𝑋 ≤ 3
d. 𝑃 2.4 ≤ 𝑋 < 4.5
2. M is a uniformly distributed continuous random variable such that 𝑀~𝑅 −1.3, 2.5 .
Find:
a. The probability density function of M.
b. The mean of M
c. The variance of 2M
d. 𝑃 0 < 𝑀 ≤ 1