The document provides an overview of Bernoulli distribution, a discrete probability distribution representing a random variable that takes the value 1 with probability p and 0 with probability 1-p. It discusses properties such as the probability mass function, moments, arithmetic mean, variance, and its application in modeling binary outcomes in experiments and logistic regression. Additionally, it notes that Bernoulli distribution serves as a special case of the binomial distribution with n=1.

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2. Bernoulli Distribution
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3. Bernoulli Distribution
The Bernoulli distribution, named after Swiss
mathematician Jacob Bernoulli, is a discrete
probability distribution of a random variable
which takes the value 1 with probability p and
the value 0 with probability 1-p.
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4. Bernoulli Distribution
X is a discrete r.v with values
X = 1 with prob. ‘p’ and X = 0 with prob. (1-p)
ie, f(X=1) = p
f(X=0) = 1-p, X =0,1
f(x) = px(1-p)1-x
let q = 1-p then
Pmf of X is f(x) = pxq1-x, x = 0,1
There fore f(x) = pxq1-x, x = 0,1
= 0 elsewhere
p is the parameter of Bernoulli distribution, q=1-p &
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5. Bernoulli Distribution
Distribution function F(X) =
F(X) =
The Bernoulli distribution is a special case of
the binomial distribution with n = 1
The Bernoulli distribution is simply B( 1,p)
i.e X Bernoulli (p)
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6. Bernoulli Distribution
r th raw moment about origin
= E(Xr ) =
here x takes the values 0 & 1 & f(x) = pxq1-x
= E(Xr ) =
= 0+ p q0
= p
Irrespective of the grade of raw moment its is “p” in Bernoulli
distribution
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7. Bernoulli Distribution
Direct way to calculate raw moments
= E(X) =
=
= 0+ 1 p
= p
= E(X2 ) =
=
= 0+12 p
= p
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8. Bernoulli Distribution
Arithmetic Mean
= E(X) = p
Variance
V(X) = E(X2 ) –[ E(X)]2
= p - p2 = p(1-p)= pq
Std.deviation
=
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9. Bernoulli Distribution
Moment generating function
Mx(t) = E( ) =
=
=
=
= q + p
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10. Bernoulli Distribution
Characteristic function
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11. Bernoulli Distribution
If X Bernoulli (p)
pmf f(x) = pxq1-x, x = 0,1,
= 0 elsewhere
p is the parameter of Bernoulli distribution, p+q=1
Arithematic Mean = p
Variance pq
m.g.f Mx(t) =q + p
Ch.fun
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12. Bernoulli Distribution
Graph of pmf of Bernoulli
Distribution
Table of pmf of Bernoulli
distribution
f(0) f(1)
P =0.5 .5 .5
P=0.4 .6 .4
P=0.6 .4 .6
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0
0.1
0.2
0.3
0.4
0.5
0.6
P =0.5
f(0)
f(1)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
P=0.4
f(0)
f(1)
0
0.2
0.4
0.6
0.8
P=0.6
f(0)
f(1)

13. Bernoulli Distribution
In experiments and clinical trials, the Bernoulli distribution is
sometimes used to model a single individual experiencing an
event like death, a disease, or disease exposure.
The model is and excellent indicator of the probability a person
has the event in yes/ no question.
Bernoulli distributions are used in logistic regression to model
disease occurrence.
It is also a special case of the two-point distribution, for which
the possible outcomes need not be 0 and 1.
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Suchithra’s Statistics classes
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