---
title: "Expectation of discrete variable; P(X≥4)"
subject: "Probability and Statistics"
chapter: 2
key: "semester_5/probability_and_statistics:expectation-discrete-rv-table"
source: https://www.bctengineer.com/q/semester_5/probability_and_statistics/expectation-discrete-rv-table
site: BCT Engineering
---

# Expectation of discrete variable; P(X≥4)

Probability and Statistics · Chapter 2: Discrete Probability Distribution

- Asked in: 2072 Chaitra, 5 marks
- Video: [Discrete Random Variables   The Expected Value of X and VarX](https://www.youtube.com/watch?v=9Q1V9kZrlsE) — Maths Genie
- Answer checked by Claude Sonnet 5.5 on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_5/probability_and_statistics/expectation-discrete-rv-table
- Whole chapter: https://www.bctengineer.com/md/semester_5/probability_and_statistics/chapter-2

## Question

Define mathematical expectation of a discrete random variable. A probability distribution is given.
| X=x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| p(X=x) | 0.26 | 0.25 | 0.11 | 0.02 | 0.25 | 0.11 |
Find (a) P(X ≥ 4); (b) P(0 < X < 4); (c) mean and variance of X.

## Short answer

Expectation is the probability-weighted average. From the table: P(X≥4)=0.36, P(0<X<4)=0.38, mean=2.08, variance=3.2936.

## Answer

#### Mathematical expectation
For a discrete random variable X with pmf p(x), the **mathematical expectation** (mean) is the probability-weighted average of its values:

E(X) = Σ x · p(x)

and E[g(X)] = Σ g(x) p(x). It is the long-run average value of X.

The given probabilities add to 0.26 + 0.25 + 0.11 + 0.02 + 0.25 + 0.11 = 1, so it is a valid distribution.

#### (a) P(X ≥ 4)
P(X ≥ 4) = P(4) + P(5) = 0.25 + 0.11 = **0.36**

#### (b) P(0 < X < 4)
P(1) + P(2) + P(3) = 0.25 + 0.11 + 0.02 = **0.38**

#### (c) Mean and variance
```text
x   p(x)    x·p(x)   x²·p(x)
0   0.26    0.00     0.00
1   0.25    0.25     0.25
2   0.11    0.22     0.44
3   0.02    0.06     0.18
4   0.25    1.00     4.00
5   0.11    0.55     2.75
Sum 1.00    2.08     7.62
```
Mean = E(X) = **2.08**

Variance = E(X²) − [E(X)]² = 7.62 − (2.08)² = 7.62 − 4.3264 = **3.2936**

**Answer: (a) 0.36, (b) 0.38, (c) mean = 2.08, variance = 3.2936**

## Check yourself

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

1. The mathematical expectation of a discrete X is:
   A. Sum of p(x)
   B. The most probable value
   C. Sum of x p(x), the probability-weighted average
   D. The simple average of the x values
   Answer: C. E(X) = sum x p(x), the long-run average. (Key point: Expectation)

2. P(X >= 4) is:
   A. 0.25
   B. 0.38
   C. 0.36
   D. 0.64
   Answer: C. P(4) + P(5) = 0.25 + 0.11. (Key point: P(X>=4))

3. P(0 < X < 4) includes:
   A. x = 1, 2, 3 only, giving 0.38
   B. x = 0 to 4, giving 0.89
   C. x = 1 to 4, giving 0.63
   D. x = 2 only
   Answer: A. The inequalities are strict, so 0 and 4 are excluded. (Key point: P(0<X<4))

4. Variance is:
   A. 7.62 - 2.08 = 5.54
   B. 2.08^2 = 4.3264
   C. 7.62 + 4.3264
   D. 7.62 - 2.08^2 = 3.2936
   Answer: D. Var = E(X^2) - [E(X)]^2. (Key point: Variance)

## Related in Probability and Statistics

- [Discrete and continuous random variables](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Adiscrete-continuous-rv-mean-variance) — Probability and Statistics, chapter 2
- [Continuous random variable; pdf properties](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Acontinuous-rv-pdf-properties) — Probability and Statistics, chapter 3
- [Breakdown voltage: normal, variance 2.25](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Abv-39-42-lt44-gt43) — Probability and Statistics, chapter 3, asked 3×
- [Discrete distribution; compare NB and binomial](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Adiscrete-distribution-nb-vs-binomial) — Probability and Statistics, chapter 2
- [Exponential cdf: P(X>2), mean, variance](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Aexponential-cdf-1-minus-e-2x) — Probability and Statistics, chapter 3, asked 3×
- [pdf kx³(4−x)²: k, mean, variance](https://www.bctengineer.com/question.md?key=semester_5%2Fprobability_and_statistics%3Apdf-kx3-4-minus-x-squared) — Probability and Statistics, chapter 3, asked 2×

## Related in other subjects

- [G-S two iterations, lines 0.02+j0.11](https://www.bctengineer.com/question.md?key=semester_5%2Fpower_system_analysis_ii%3Ags-two-iterations-2080) — Power System Analysis II, chapter 2
- [Moments of random variable; NEB of RC-LPF; output noise power](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Amoments-neb-rc-lpf) — Communication Systems, chapter 9
- [Hardness of well water with strontium 25.26 mg/l](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Ahardness-bk81) — Water Supply Engineering, chapter 4
- [Settling tank 30 m x 3 m, 0.02 mm at 25°C; 0.01 mm at 10°C](https://www.bctengineer.com/question.md?key=semester_5%2Fwater_supply_engineering%3Ased-as70) — Water Supply Engineering, chapter 6, asked 3×