RD Sharma Class 12 Solutions Chapter 32 - Mean and Variance of a Random Variable (Ex 32.1) Exercise 32.1 - Free PDF
FAQs on RD Sharma Class 12 Solutions Chapter 32 - Mean and Variance of a Random Variable (Ex 32.1) Exercise 32.1
1. Where can I find accurate, step-by-step solutions for all questions in RD Sharma Class 12 Maths Chapter 32, Exercise 32.1?
You can find detailed, step-by-step solutions for every problem in RD Sharma Class 12 Maths Chapter 32, Exercise 32.1, on Vedantu. These solutions are prepared by subject matter experts and are aligned with the latest CBSE guidelines for the 2025-26 academic session, ensuring you understand the correct method for finding the mean and variance of a random variable.
2. What is the first step to solve a problem that asks for the mean and variance of a random variable in Exercise 32.1?
The first and most crucial step is to determine the probability distribution of the random variable X. This involves:
- Identifying all possible values that the random variable X can take (x₁, x₂, x₃, ...).
- Calculating the probability P(X=xᵢ) for each of these values.
- Organising these values and their corresponding probabilities in a table format.
3. How do you calculate the mean, or expected value E(X), of a discrete random variable?
The mean or expected value E(X) of a discrete random variable is calculated by summing the product of each possible value of the random variable and its corresponding probability. The formula is: E(X) = Σxᵢpᵢ. In the context of Exercise 32.1, you would multiply each value 'x' by its probability 'p' and add all these products together.
4. What does the 'mean' of a random variable actually represent? Is it the same as a simple average?
The mean of a random variable, also known as the expected value, represents the long-term average outcome of a random experiment if it were repeated many times. It is a weighted average, where each possible value is weighted by its probability of occurrence. This is different from a simple average, where all values are given equal weight.
5. How is the variance of a random variable calculated, and what does it tell us?
The variance, denoted as Var(X) or σ², measures the spread or dispersion of the random variable's values around its mean. A low variance indicates that the values are clustered close to the mean, while a high variance indicates they are more spread out. The most common formula used in this chapter is: Var(X) = E(X²) - [E(X)]², where E(X²) = Σxᵢ²pᵢ.
6. What is a common mistake to avoid when calculating the variance using the formula Var(X) = E(X²) - [E(X)]²?
A very common mistake is confusing E(X²) with [E(X)]².
- To find E(X²), you must first square each value of the random variable (xᵢ²), then multiply by its corresponding probability (pᵢ), and finally sum up all these products (Σxᵢ²pᵢ).
- [E(X)]² is simply the square of the mean that you have already calculated.
7. Why must the sum of all probabilities in a probability distribution table always equal 1?
The sum of all probabilities, ΣP(X=xᵢ), must equal 1 because the set of all possible values {x₁, x₂, x₃, ...} represents all possible outcomes of the experiment. Since it is certain that one of these outcomes will occur, the total probability must be 1. This principle is often used in problems to find an unknown probability 'k' in a given distribution.
