this is ppt for probability proficiency

1. M
Madhav Institute of Technology & Science, Gwalior (M.P.)
A Govt. Aided UGC Autonomous & NAAC Accredited Institute Affiliated to R.G.P.V., Bhopal
Department of Engineering Mathematics & Computing
Probability and Random Processes
Artificial Intelligence & Data Science Branch
Submitted To:
DR. DIVYA CHATURVEDI
Assistant Professor
Submitted By:
DIVYANSH KHARE - 0901AD211013, DRISHTI JAIN - 0901AD211014,
GAURAV CHAUDHARY - 0901AD211015, GIREESH KOSHE - 0901AD211016,
HARSH GUPTA - 0901AD211017, HARSH PATEL - 0901AD211018

2. Probability
Density Function
Properties of
Probability
Density
Functiom
Properties of
Distribution
Function
Normal
Distribution
Distribution
Function
Parameters of
The Normal
Distribution
01
04
02
05
03
06
Table of contents
07
Exponential
Distribution
Central Limit
Theorem
Distribution
Function of
Exponential Function
09
08

3. Probability Density Function
Let the probability that the continuous random variable X lies within
infinitesimal interval (𝑥 −
𝑑𝑥
2
, 𝑥 +
𝑑𝑥
2
) is
𝑃 𝑥 −
𝑑𝑥
2
≤ 𝑋 ≤ 𝑥 +
𝑑𝑥
2
= 𝑓 𝑥 . 𝑑𝑥
Where 𝑓 𝑥 is a continuous function of 𝑥, is called the Probability Density
Function and the curve 𝑦 = 𝑓 𝑥 is called Probability Curve.
Above equation can also be written as,
𝑃 𝑥 −
𝑑𝑥
2
≤ 𝑋 ≤ 𝑥 +
𝑑𝑥
2
𝑑𝑥
= 𝑓(𝑥)
⇒ lim
𝛿𝑥→0
𝑃 𝑥 −
𝑑𝑥
2
≤ 𝑋 ≤ 𝑥 +
𝑑𝑥
2
𝛿𝑥
= 𝑓(𝑥)

4. 0
𝑥 −
𝑑𝑥
2
𝑥 +
𝑑𝑥
2
𝑑𝑥
𝑓(𝑥)
𝑦 = 𝑓(𝑥)
𝑃 𝑥 −
𝑑𝑥
2
≤ 𝑋 ≤ 𝑥 +
𝑑𝑥
2
= 𝑓 𝑥 . 𝑑𝑥
𝑥
𝑦
𝑎 𝑏
𝑃 𝑎 ≤ 𝑋 ≤ 𝑏 =
𝑎
𝑏
𝑓 𝑥 . 𝑑𝑥

5. Properties of Probability
Density Function
• If the value of continuous random variable 𝑋𝜖[𝛼, 𝛽] then value of
total probability 𝑃 𝛼 ≤ 𝑋 ≤ 𝛽 = 𝛼
𝛽
𝑓 𝑥 . 𝑑𝑥 = 1 continuous
random variable 𝑋1 represents total probability for all the possible
values of 𝑋.
• If 𝛼 = −∞ and 𝛽 = +∞,
−∞
∞
𝑓 𝑥 . 𝑑𝑥 = 1
• 𝑓 𝑥 ≥ 0
• For finding probability at a point 𝑃 𝑋 = 𝑐 ,
𝑃 𝑐 ≤ 𝑋 ≤ 𝑥 =
𝑐
𝑐
𝑓 𝑥 . 𝑑𝑥 = 0
𝑖. 𝑒. 𝑃 𝑋 = 𝑐 = 0

6. Distribution Function
Let 𝑋 be a random variable. The function 𝐹
defined for all real 𝑥 by 𝐹 𝑥 = 𝑃 𝑋 ≤ 𝑥 =
𝑃 𝜔: 𝑋 𝜔 ≤ 𝑥 , −∞ < 𝑥 < ∞ is called
distribution function of the random variable 𝑋.
A distribution is also called the cumulative
distribution function. It is also denoted by 𝐹𝑋 𝑥 .
Domain of distribution function = (−∞, ∞)
& Range of distribution function = [0,1]

7. Properties of
Distribution Function
• If 𝐹 is the distribution function of the random
variable 𝑋 and if 𝑎 < 𝑏, then 𝑃 𝑎 < 𝑋 < 𝑏 =
𝐹 𝑏 − 𝐹(𝑎).
• 0 ≤ 𝐹 𝑥 ≤ 1: ∀ 𝑥 ∈ −∞, ∞
𝑥 < 𝑦 ⇒ 𝐹 𝑥 ≤ 𝐹(𝑦)
• 𝐹 −∞ = 0
𝐹 ∞ = 1

8. Example-
Suppose if we toss 3 coins together then sample space 𝑆 =
{𝐻𝐻𝐻, 𝐻𝐻𝑇, 𝐻𝑇𝑇, 𝑇𝑇𝑇, 𝑇𝑇𝐻, 𝑇𝐻𝐻, 𝑇𝐻𝑇, 𝐻𝑇𝐻} and discrete random
variable 𝑋 denotes number of head. Find distribution function for the
event.
Solution-
𝑋 = 𝑛𝑜. 𝑜𝑓 ℎ𝑒𝑎𝑑𝑠
𝑋 = 0,1,2,3
Then probability for all random variable 𝑋
𝑃 𝑋 = 0 = 𝑃 𝑇𝑇𝑇 =
1
8
𝑃 𝑋 = 1 = 𝑃 𝐻𝑇𝑇 𝑜𝑟 𝑇𝑇𝐻 𝑜𝑟 𝑇𝐻𝑇 =
3
8
𝑃 𝑋 = 2 = 𝑃 𝐻𝐻𝑇 𝑜𝑟 𝑇𝐻𝐻 𝑜𝑟 𝐻𝑇𝐻 =
3
8
𝑃 𝑋 = 3 = 𝑃 𝐻𝐻𝐻 =
1
8

9. 𝑋 0 1 2 3
𝑃 𝑋 = 𝑥 1
8
3
8
3
8
1
8
𝐹 0 = 𝑃 𝑋 ≤ 0 = 𝑃 𝑋 = 0 =
1
8
𝐹 1 = 𝑃 𝑋 ≤ 1 = 𝑃 𝑋 = 0 + 𝑃 𝑋 = 1 =
1
2
𝐹 2 = 𝑃 𝑋 ≤ 2 + 𝑃 𝑋 = 0 + 𝑃 𝑋 = 1 + 𝑃 𝑋 = 2 =
7
8
𝐹 3 = 𝑃 𝑋 ≤ 3 = 𝑃 𝑋 = 0 + 𝑃 𝑋 = 1 + 𝑃 𝑋 = 2 + 𝑃 𝑋 = 3 = 1
𝐹 4 = 𝑃 𝑋 ≤ 4 = 𝑃 𝑋 = 0 + 𝑃 𝑋 = 1 + 𝑃 𝑋 = 2 + 𝑃 𝑋 = 3 + 𝑃 𝑋 = 4 = 1
𝐹 5 = 1
𝐹 6 = 1
.
.
.

10. 𝐹 𝑥 =
0; 𝑥 < 0
1
8
; 0 ≤ 𝑥 ≤ 1
1
2
; 1 ≤ 𝑥 < 2
7
8
; 2 ≤ 𝑥 < 3
1; 𝑥 ≥ 3
𝑥 < 𝑦 ⇒ 𝐹 𝑥 ≤ 𝐹 𝑦 {Increasing Function}
Distribution Function is an increasing function.

11. Normal Distribution
The normal distribution is a continuous probability distribution that is symmetrical
around its mean, most of the observations cluster around the central peak, and the
probabilities for values further away from the mean taper off equally in both
directions. Extreme values in both tails of the distribution are similarly unlikely.
While the normal distribution is symmetrical, not all symmetrical distributions are
normal.
The Normal (or Gaussian) Distribution is defined by the probability density function
𝑓 𝑥 =
1
𝜎 2𝜋
𝑒−1/2(
𝑥−𝜇
𝜎
)2
, 𝑓𝑜𝑟 − ∞ ≤ 𝑥 ≤ ∞
where 𝜇 and 𝜎 > 0 are the parameters of distribution.
Clearly 𝑓 𝑥 is non-negative and −∞
∞
𝑓 𝑥 . 𝑑𝑥 = 1.
Demoivre made the discovery of this distribution in 1773.

12. Parameters of The Normal
Distribution
Mean (𝜇)
The mean is the central tendency of the normal distribution. It defines
the location of the peak for the bell curve. Most values cluster
around the mean. On a graph, changing the mean shifts the entire
curve left or right on the X-axis.
Standard deviation (𝜎)
The standard deviation is a measure of variability. It defines the width of
the normal distribution. The standard deviation determines how far
away from the mean the values tend to fall. It represents the typical
distance between the observations and the average.

13. On a graph, changing the standard deviation either tightens or spreads
out the width of the distribution along the X-axis. Larger standard
deviations produce wider distributions.
Variance (𝜎2
)
Median 𝑚 = 𝜇
The median of 𝑋 is the same as its mean.
Mode =
1
𝜎 2𝜋

14. Exponential Distribution
The probability distribution having the probability density function 𝑓(𝑥), defined by
𝑓 ∞ =
0, 𝑓𝑜𝑟 𝑥 ≤ 0
𝜃𝑒−𝜃𝑥
, 𝑓𝑜𝑟 𝑥 ≥ 0, 𝜃 > 0
is called exponential distribution with parameter 𝜃.
Here 𝑓 ∞ = −∞
∞
𝑓 𝑥 . 𝑑𝑥 = −∞
0
𝑓 𝑥 . 𝑑𝑥 + 0
∞
𝑓 𝑥 . 𝑑𝑥
= 0 +
0
∞
𝜃𝑒−𝜃𝑥
𝑑𝑥
= [−𝑒−𝜃𝑥
]0
∞
= 1

15. Distribution Function of
Exponential Distribution
𝐹 𝑥 = 𝑃 𝑋 ≤ 𝑥 =
−∞
𝑥
𝜃𝑒−𝜃𝑥
𝑑𝑥
⇒ 𝐹 𝑥 =
0, 𝑥 < 0
1 − 𝑒−𝜃𝑥
, 𝑥 ≥ 0
1, 𝑥 = ∞
Sometimes exponential distribution is defined by the p.d.f.
𝑓 𝑥 =
1
𝛽
𝑒
−
1
𝛽
𝑥
, 𝑥 > 0
= 0, otherwise.

16. Central Limit Theorem
In probability theory, the central limit theorem (CLT) establishes that, in many
situations, when independent random variables are summed up, their
properly normalized sum tends toward a normal distribution even if the original
variables themselves are not normally distributed.
Let 𝑋1, 𝑋2, … … , 𝑋𝑛 be 𝑛 independent random variables all of which have the same
distribution. Let the common expectation and variance be 𝜇 and 𝜎2
respectively. Let
𝑋 = 𝑖=1
𝑛
𝑋𝑖/𝑛. Then the distribution of 𝑋 approaches the normal distribution with
mean 𝜇 and variance 𝜎2
/𝑛 as 𝑛 → ∞ 𝑖. 𝑒. variate 𝑍 = (𝑋 − 𝜇)/(
𝜎
𝑛
) has standard normal
distribution.

17. For Your Attention.
Thank You !!!