Download Mean variance Moments

MA305
Mean, Variance
Binomial Distribution
By: Prof. Nutan Patel
Asst. Professor in Mathematics
IT-NU
A-203
patelnutan.wordpress.com
MA305 Mathematics for ICE
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Probability Function
If for random variable X, the real valued function f(x) is such that
P(X=x) = f(x)
Then f(x) is called probability function.
If X is a discrete random variable then its probability function f(x) is
discrete probability function. It is called probability mass function.
 f x   1
i
If X is a continuous random variable then its probability function f(x)
is continuous probability function. It is called probability density
function.
 f x  dx  1


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Expected value
Definition: If X is a random variable with values x1, x2, …, xn and
corresponding probabilities p1, p2, …, pn then the expected value of X,
denoted as E(X) is
E(X)= p1 x1 + p2 x2 + … + pn xn .
 Expectation
The mean value (µ) of the probability distribution of a variable X is commonly
known as its expectation and is denoted by E(X).
E(X)= 𝑛𝑖=1 xi 𝑓(xi)
(Discrete Distribution)
∞
E(X)= −∞ 𝑥 𝑓(𝑥)
(Continuous Distribution)
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MEAN
The mean value (µ) of the probability distribution of a variable X is commonly
known as its expectation and is denoted by E(X).
E(X)= 𝑛𝑖=1 xi 𝑓(xi)
(Discrete Distribution)
∞
E(X)= −∞ 𝑥 𝑓(𝑥)
(Continuous Distribution)
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Variance
• Variance characterizes the variability in the distributions, since two
distributions with same mean can still have different dispersion of
data about their means.
• Variance of R.V. X is
 2  E x   2   xi   2 f xi 
 2  E  x   2 

2


X


f  x dx


• Remark:
 2  E X 2   E  X 2  E X 2    2
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Standard Deviation (S.D.)
• S.D. Denoted by , is the positive square root of variance.
Ex: Prove that
(a) E(cX)=c E(X) (b) E(X+c) =E(X)+c
Ex: Prove that
(a) Var(cX) = c2 Var(X)
(b) Var(X+c)= Var(X)
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Moment Generating Function (mgf)
• MGF, the expected value of 𝑒 𝑡𝑥 for the probability distribution
function f(x) of a random variable X
m. g. f. is defined as 𝐸 𝑒 𝑡𝑥 = 𝑀0 𝑡 = 𝑀 𝑡 = 𝑒 𝑡𝑥 𝑓(𝑥)
OR
Series form of m.g.f.
∞ 𝑡𝑥
𝑀 𝑡 = −∞ 𝑒 𝑓(𝑥)
2
3






tx
tx
tx

M 0 (t )  E e   E 1  tx 

 
2!
3!


2
3
2 t
3 t
 1  E  x t  E x   E x   
2!
3!
t2
t3
 1  1t   2
  3

2!
3! r
Where is  r the usual moment of order r about the origin, is the coefficient of
MA305 Mathematics for ICE
t
.
r!
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• M.g.f. of the distribution about any other value x=a is defined as

 

M a (t )  E e t  xa   E e tx e  at  e  at E (e tx )  e  at M 0 t 
• Moments about mean is known as central moments.
•
Mean       .
1
1
Variance  2   2  1
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Bernoulli Trail Experiments
• Suppose that you toss a coin ten times. What is the probability that
heads appears seven out of ten times?
• A student guesses at all the answer on a ten MCQs quiz.
Such problems involved repeated trials of an experiment with only two
possible outcomes: heads or tails, right or wrong, win or loss, so on..
• We classify the two outcomes as success or failure.
• When outcomes of an experiment are divided into two parts, it is
called the situation of dichotomy.
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• Properties of Bernoulli Experiment
1. The experiment is repeated a fixed number of times (n times).
2. Each trial has only two possible outcomes: success and failure. The
outcomes are exactly the same for each trial.
3. The probability of success remains the same for each trial.
(probability of success is p and probability of failure q=1-p).
4. The trials are independent.
5. We are interested in the total number of successes, not the order in
which they occur.
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• Probability of a Bernoulli experiment:
The probability of r successes in n trials is
𝑛 𝑟 𝑛−𝑟
P(r successes in n trials)=P(r)=
𝑝 𝑞
𝑟
where n is independent trials,
p is probability of success in a single trial,
q=1-p is the probability of failure in a single trial,
r is the number of successes (r ≤ n).
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Ex: A coin is tossed ten times. What is the probability that heads occurs
seven times?
Ans: here, n=10, r=7, p=1/2, q=1/2.
𝑛 𝑟 𝑛−𝑟
P(r=7)=
𝑝 𝑞
𝑟
10 1 7 1 3
=
2
7 2
= 0.11718
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Ex: Find the probability of 4 successes, where n=9 and p=0.4.
Ans: 0.15049
Ex: Determine the probability of getting sum 9 exactly 2 times in 3 throws with a
pair of dice.
Ans: n=3, r=2,
sum is 9={(3, 6), (4, 5), (5,4), (6, 3)},
p=probability of getting sum 9= 4/36=1/9.
q=1-p= 8/9.
3 1 2 8 1
P(r=2)=
=8/243=0.0329
9
9
2
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Binomial Distribution
• Bernoulli experiment trials distribution is called a binomial
distribution
• For each integer value r, 0≤r≤n, find the probability of r successes in n
trials. Then the distribution obtained is the binomial probability
distribution.
• This distribution is discrete distribution.
• Definition:
Let X be the random variable for a binomial distribution with n
repeated trials, with p the probability of success, q the probability of
𝑛 𝑟 𝑛−𝑟
failure and P(X=r)=
𝑝 𝑞
, r= 0, 1, 2, 3,…, n.
𝑟
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Application of Binomial Distribution
1. In Quality control charts (fraction defective or number of defective
per sample)
2. Useful for insurance companies.
3. It is very useful in the application pertaining to behavioural sciences.
4. In research field where dichotomy is there, this distribution is used.
5. Estimation of reliability of systems.
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Ex: Find the binomial distribution for n=4 and p=0.3.
X
0
1
2
3
4
P(X=x)
4
0
4
1
4
2
4
3
4
4
P(X)
(0.3)0 (0.7)4
(0.3)1 (0.7)3
(0.3)2 (0.7)2
(0.3)3 (0.7)1
(0.3)4 (0.7)0
• 0.2401
• 0.4116
• 0.2646
• 0.0756
• 0.0081
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Mean, Variance of Binomial Distribution
Mean = np.
Variance = npq.
Standard Deviation = 𝑛𝑝𝑞
Ex: Find mean and s.d. of the binomial distribution with n=20 and
p=0.35.
Ans: mean= 7, variance= 4.55, s.d. =2.133
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• EX: Form the binomial distribution of the experiment of tossing a coin
six times and counting the number of heads.
EX: Compute mean, variance and s.d. of followings
1. n=50, p=0.4
2. N=600, p=0.52
3. N=470, p=0.08
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• EX: If 10% of the rivets produced by a machine are defective, find the
probability that out of 5 rivets chosen at random (i) none will be
defective, (ii) one will be defective, and (iii) at least two will be
difective.
• Ans: 0.5905, 0.32805, 0.08146.
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