Unit 4: Probability Distributions

Introduction

A random variable is a real-valued function whose domain is the sample space of a random experiment. It can be discrete (taking countable values) or continuous (taking uncountably many values in an interval).

Mathematical Expectation

The expected value representing the weighted average of all possible values. For a discrete random variable \( X \):

\[ E(X) = \sum x_i P(X = x_i) \]

The variance, measuring the spread of the data, is given by:

\[ \operatorname{Var}(X) = E(X^2) - [E(X)]^2 \]

Important Distributions

1. Binomial Distribution

A distribution resulting from \( n \) independent Bernoulli trials, each with probability of success \( p \).

\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \] where \( q = 1 - p \).

• Mean = \( np \)
• Variance = \( npq \)

2. Poisson Distribution

Applied to processes where events occur randomly in a continuous space or time. Used when \( n \) is very large and \( p \) is very small.

\[ P(X = x) = \frac{\lambda^x e^{-\lambda}}{x!} \] where \( \lambda \) is the mean number of occurrences.

3. Normal Distribution

A continuous symmetric distribution forming a classic bell curve.

The standard normal variate is defined as: \[ Z = \frac{X - \mu}{\sigma} \]

Competency-Based Questions

Q1. The probability of predicting the correct outcome of a match is 0.4. If a sports analyst predicts 10 matches, find the probability of exactly 6 correct predictions using the Binomial Distribution formula.

Q2. An insurance call center receives on average 3 calls per minute. Apply the Poisson distribution to determine the probability that in any given minute, they will receive exactly 5 calls. (Given \( e^{-3} = 0.0498 \))

Q3. The scores of a competitive exam form a normal distribution with a mean of 60 and a standard deviation of 12. Determine the percentage of candidates who scored above 84. (Assume standard normal table values for \( Z \)).

Q4. Let \( X \) be a discrete random variable signifying the number of defective computers in a batch of 4. Calculate the expected value \( E(X) \) and variance of the defectives if the probability of encountering a defective computer is 10%.

Answers

Ans 1. \( n=10, p=0.4, q=0.6, r=6 \). Probability \( P(X=6) = \binom{10}{6} (0.4)^6 (0.6)^4 pprox 0.111 \).

Ans 2. \( \lambda=3, x=5 \). \( P(X=5) = \frac{3^5 e^{-3}}{5!} = \frac{243 \times 0.0498}{120} pprox 0.1008 \).

Ans 3. \( Z = \frac{84 - 60}{12} = 2 \). The area to the right of \( Z = 2 \) is \( 0.5 - 0.4772 = 0.0228 \). Hence, 2.28% candidates scored above 84.

Ans 4. Since it is a Binomial setting with \( n=4, p=0.1 \), \( E(X) = np = 4 \times 0.1 = 0.4 \) and Variance \( = npq = 4 \times 0.1 \times 0.9 = 0.36 \).