Lesson 1.5: Probability & Statistics (Distributions, Mean, Variance, Correlation)

Probability & Statistics are essential for engineering problem-solving and aptitude. GATE and PSU exams often include questions on probability, random variables, distributions, central tendency, dispersion, and correlation.

🔹 1. Probability

Definition:
Probability of an event $A$ is:

P(A)=Number of favorable outcomesTotal number of outcomesP(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}P(A)=Total number of outcomesNumber of favorable outcomes​

Laws of Probability:

• $0\le P(A)\le 1$

• $P(\text{sure event})=1$, $P(\text{impossible event})=0$

• Addition Rule: $P(A\cup B)=P(A)+P(B)-P(A\cap B)$

• Conditional Probability: P(A∣B)=P(A∩B)P(B)P(A|B) = \frac{P(A \cap B)}{P(B)}P(A∣B)=P(B)P(A∩B)​

• Multiplication Rule: $P(A\cap B)=P(A)P(B|A)$

Example:
A die is rolled. Probability of getting an even number = 3/6 = 1/2

🔹 2. Random Variables & Distributions

• Random Variable: A variable whose values are outcomes of a random experiment.

  • Discrete: finite/countable values

  • Continuous: any value in an interval

• Common Distributions in GATE/PSU:

  • Binomial: P(X=k)=(nk)pk(1−p)n−kP(X=k) = \binom{n}{k} p^k (1-p)^{n-k}P(X=k)=(kn​)pk(1−p)n−k

  • Poisson: P(X=k)=λke−λk!P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}P(X=k)=k!λke−λ​

  • Normal (Gaussian): f(x)=1σ2πe−(x−μ)22σ2f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}f(x)=σ2π​1​e−2σ2(x−μ)2​

Applications:

• Modeling defects in manufacturing

• Reliability analysis

• Measurement errors

🔹 3. Mean, Variance, Standard Deviation

• Mean (Expected Value):

μ=E[X]=∑xiP(xi)or∫xf(x)dx\mu = E[X] = \sum x_i P(x_i) \quad \text{or} \quad \int x f(x) dxμ=E[X]=∑xi​P(xi​)or∫xf(x)dx

• Variance:

σ2=E[(X−μ)2]=∑(xi−μ)2P(xi)\sigma^2 = E[(X-\mu)^2] = \sum (x_i-\mu)^2 P(x_i)σ2=E[(X−μ)2]=∑(xi​−μ)2P(xi​)

• Standard Deviation:
  σ=σ2\sigma = \sqrt{\sigma^2}σ=σ2​

Example:
Discrete variable X = {1, 2, 3}, P(X=1)=0.2, P(X=2)=0.5, P(X=3)=0.3

• Mean: $\mu =1\ast 0.2+2\ast 0.5+3\ast 0.3=2.1$

• Variance: σ2=(1−2.1)2∗0.2+(2−2.1)2∗0.5+(3−2.1)2∗0.3=0.49\sigma^2 = (1-2.1)^2*0.2 + (2-2.1)^2*0.5 + (3-2.1)^2*0.3 = 0.49σ2=(1−2.1)2∗0.2+(2−2.1)2∗0.5+(3−2.1)2∗0.3=0.49

🔹 4. Correlation & Regression

• Correlation Coefficient (r): Measures strength & direction of linear relationship:

r=Cov(X,Y)σXσY,−1≤r≤1r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y}, \quad -1 \le r \le 1r=σX​σY​Cov(X,Y)​,−1≤r≤1

• Regression Lines: Predict value of one variable based on another:

$$Y=a+bX\text{or}X=c+dY$$

Applications:

• Quality control → correlation of temperature vs. pressure

• Engineering design → relation between load & deflection

🔹 5. Solved Examples (PYQ Style)

Example 1 (GATE ME 2016):
X = {0,1,2}, P(X=0)=0.2, P(X=1)=0.5, P(X=2)=0.3. Find mean & variance.
👉 Solution: Mean = 1.1, Variance = 0.49

Example 2 (PSU Exam):
If r = 0.8 between X (stress) and Y (strain), interpret the relationship.
👉 Solution: Strong positive correlation → as stress increases, strain increases.

🔹 6. Practice Exercises

1. A coin is tossed 3 times. Probability of exactly 2 heads.

2. Find mean & standard deviation for X = {2,4,6}, P(X) = {0.3, 0.4, 0.3}.

3. Compute correlation coefficient if Cov(X,Y)=4, σX=2,σY=3\sigma_X = 2, \sigma_Y = 3σX​=2,σY​=3.

4. Probability of at most 1 defective item in 5 items if p=0.1 (Binomial).

🔹 7. Summary

• Probability: Likelihood of events, conditional & joint probability.

• Random Variables & Distributions: Discrete/Continuous, Binomial, Poisson, Normal.

• Mean & Variance: Central tendency & dispersion.

• Correlation: Linear relationship between two variables.

• Applications: Reliability, quality control, GATE/PSU aptitude questions.