---
title: "pmf with unknown K: mean, variance"
subject: "Probability and Statistics"
chapter: 2
key: "semester_5/probability_and_statistics:pmf-with-k-mean-variance"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/pmf-with-k-mean-variance
site: BCT Engineering
---

# pmf with unknown K: mean, variance

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2076 Asoj, 6 marks
- Video: [Random Variables & Distribution Functions-To find mean n Variance](https://www.youtube.com/watch?v=gUdeh-Z4dYA) — Btech Maths Hub
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/pmf-with-k-mean-variance
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

A random variable X has following probability function:
| X | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| P(X) | K | 0.1 | 0.2 | 2k | 3k | 0.1 |
i) Find the value of K.
ii) Find Mean and Variance.

## Short answer

Sum of probabilities gives 6K+0.4=1, so K=0.1. Mean E(X)=0.8, E(X^2)=2.8; variance = 2.8-0.8^2 = 2.16.

## Answer

#### i) Value of K
The probabilities of a pmf must add up to 1:

K + 0.1 + 0.2 + 2K + 3K + 0.1 = 1
6K + 0.4 = 1 → 6K = 0.6 → **K = 0.1**

```text
X      -2    -1    0     1     2     3
P(X)   0.1   0.1   0.2   0.2   0.3   0.1
```
All values are non-negative and sum to 1.

#### ii) Mean and variance
E(X) = Σ x P(x)
= (−2)(0.1) + (−1)(0.1) + 0(0.2) + 1(0.2) + 2(0.3) + 3(0.1)
= −0.2 − 0.1 + 0 + 0.2 + 0.6 + 0.3 = **0.8**

E(X²) = Σ x² P(x)
= 4(0.1) + 1(0.1) + 0 + 1(0.2) + 4(0.3) + 9(0.1)
= 0.4 + 0.1 + 0.2 + 1.2 + 0.9 = 2.8

Variance = E(X²) − [E(X)]² = 2.8 − (0.8)² = 2.8 − 0.64 = **2.16**

**Answer: K = 0.1, mean = 0.8, variance = 2.16**

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. K is found by:
   A. Setting K equal to 0.1
   B. Setting the mean equal to 0
   C. Setting the sum of all probabilities equal to 1
   D. Setting 6K = 0.4
   Answer: C. A pmf must sum to 1: 6K + 0.4 = 1 gives K = 0.1. (Key point: Finding K)

2. What is K?
   A. 0.4
   B. 0.6
   C. 0.1
   D. 0.15
   Answer: C. 6K = 0.6 so K = 0.1. (Key point: Value of K)

3. E(X) is:
   A. 0.8
   B. 0.2
   C. 1.0
   D. 2.8
   Answer: A. Sum of x P(x) = -0.2 - 0.1 + 0 + 0.2 + 0.6 + 0.3 = 0.8. (Key point: Mean)

4. Variance is:
   A. 2.8
   B. 2.0
   C. 3.44
   D. 2.16
   Answer: D. E(X^2) = 2.8, so Var = 2.8 - 0.64 = 2.16. (Key point: Variance)

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