Statistics I · Unit 5 · 5 hrs
Random Variables and Mathematical Expectation
Exam-focused notes for Random Variables and Mathematical Expectation (Statistics I, STA169): what the TU syllabus asks and how it has actually been tested, with 4 solved past questions from this unit.
What this unit covers
- Concept of a random variable
- Types of random variables
- Probability distribution of a random variable
- Mathematical expectation of a random variable
- Addition and multiplicative theorems of expectation
- Problems and illustrative examples related to computer Science and IT
Mathematical expectation of a random variable
Define random variable. If X is the number of points rolled with a balanced die, find E(X) and variance of X. Also find the expected value of random variable g(X) = 2X^2 + 1. [5]
- Experiment: rolling a balanced (fair) die - Random variable $X$ = number of points shown - Possible values: $x = 1, 2, 3, 4, 5, 6$ - $P(X = x) = \frac{1}{6}$ for each value (balanced die) - Function required: $g(X) = 2X^2 + 1$ All data present and suffici...
Full solved answer →A coin is tossed two times and if X denotes the number of heads obtained, find (i) $E(x)$ (ii) $E(x^2)$ (iii) $V(x)$ (iv) $E(2X^2 + 3x - 5)$ [5]
- Fair coin tossed 2 times - X = number of heads - Possible values: X = 0, 1, 2 - Sample space: {TT, HT, TH, HH}, each equally likely with probability 1/4 X Outcomes P(X) ------------------- 0 TT 1/4 1 HT, TH 1/2 2 HH 1/4 Check: $\frac{1}{4}+\frac{1}{2}+\fr...
Full solved answer →The random variable $X$ has the following probability distribution.
$$\begin{array}{|c|ccccccc|}\hline X & 0 & 1 & 2 & 3 & 4 & 5 & 6 \ P(X=x) & 0.03 & 0.15 & 0.4 & 0.2 & 0.1 & 0.07 & 0.05 \ \hline \end{array}$$
[5]
X 0 1 2 3 4 5 6 ------------------------ P(X=x) 0.03 0.15 0.40 0.20 0.10 0.07 0.05 The question statement does not specify what to compute. Standard interpretation for a 5-mark problem: verify validity, then find mean, variance, and standard deviation. --- ...
Full solved answer →Probability distribution of a random variable
Write notes on any two: a. Random variable and probability distribution b. Five number summary c. Sampling error. [5]
--- Definition: The five number summary is a set of five descriptive statistical values that together provide a concise summary of a dataset. These five values are: Symbol Meaning ----------------- Xs Smallest value (minimum) Q1 First quartile (lower quarti...
Full solved answer →Make Unit 5 stick
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