Bernoulli distribution under the complementary Statistics syllabus of University of Calicut in BSc core of Mathematics, Physics & Computer Science.

1. Welcome
to
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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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