Mean of Random Variable

Probability distribution of a random variable is defined as a description accounting the values of the random variable along with the corresponding probabilities. In many cases we express the feature of random variable with the help of a single value computed from its probability distribution. These values can either be mean or median or mode.

Mean And Variance

Let

\begin{array}{l} X \end{array}

be a random variable with possible values

\begin{array}{l} x_{1}, x_{2}, x_{3}, \dots, x_{n} \end{array}

occurring with probabilities

\begin{array}{l} p_{1}, p_{2}, p_{3}, \dots, p_{n} \end{array}

, respectively. The mean of a random variable

\begin{array}{l} X \end{array}

, denoted by

\begin{array}{l} μ \end{array}

, is the weighted average of the possible values of

\begin{array}{l} X \end{array}

, each value being weighted by its probability of occurrence. The mean of a random variable X is also knows as expectation of

\begin{array}{l} X \end{array}

given by,

\begin{array}{l} E (X) = μ = \sum_{i = 1}^{n} x_{i} p_{i} \end{array}

\begin{array}{l} x_{1} p_{1} + x_{2} p_{2} + \dots + x_{n} p_{n} \end{array}

Example

Illustration 1: Calculate the mean of the number obtained on rolling an unbiased die.

Solution: The sample space of the experiment,

\begin{array}{l} S \end{array}

= {1, 2, 3, 4, 5, 6}.

Let the number obtained after rolling the die be

\begin{array}{l} X \end{array}

. Basically,

\begin{array}{l} X \end{array}

is a random variable which can take any value from 1, 2, 3, 4, 5 and 6.

\begin{array}{l} P (1) \end{array}

represents probability of getting 1 after rolling the die, then

\begin{array}{l} P (1) = P (2) = P (3) = P (4) = P (5) = P (6) = \frac{1}{6} \end{array}

Probability distribution of X can be given as,

X	1	2	3	4	5	6
P(X)	$\begin{array}{l} \frac{1}{6} \end{array}$	$\begin{array}{l} \frac{1}{6} \end{array}$	$\begin{array}{l} \frac{1}{6} \end{array}$	$\begin{array}{l} \frac{1}{6} \end{array}$	$\begin{array}{l} \frac{1}{6} \end{array}$	$\begin{array}{l} \frac{1}{6} \end{array}$

\begin{array}{l} E (X) = μ = \sum_{i = 1}^{n} x_{i} p_{i} \end{array}

\begin{array}{l} \sum_{i = 1}^{6} x_{i} p_{i} = 1. \frac{1}{6} + 2. \frac{1}{6} + 3. \frac{1}{6} + 4. \frac{1}{6} + 5. \frac{1}{6} + 6. \frac{1}{6} \end{array}

\begin{array}{l} \frac{1}{6} + \frac{2}{6} + \frac{3}{6} + \frac{4}{6} + \frac{5}{6} + \frac{6}{6} = \frac{21}{6} = 3.5 \end{array}

There is an important point to note here. If each of the values of a random variable (

\begin{array}{l} a_{1}, a_{2}, \dots, a_{n} \end{array}

) has equal probability of occurring (

\begin{array}{l} \frac{1}{n} \end{array}

), then mean is given by

\begin{array}{l} (\frac{a_{1} + a_{2} + \dots + a_{n}}{n}) \end{array}

Mean of random variables with different probability distributions can have same values. Hence, mean fails to explain the variability of values in probability distribution. Therefore, variance of random variable is defined to measure the spread and scatter in data. Variance of a random variable is discussed in detail here on.

Variance of random variable

Basically, the variance tells us how spread-out the values of X are around the mean value. Variance of a random variable (denoted by

\begin{array}{l} σ_{x}^{2} \end{array}

) with values

\begin{array}{l} x_{1}, x_{2}, x_{3}, \dots, x_{n} \end{array}

occurring with probabilities

\begin{array}{l} p_{1}, p_{2}, p_{3}, \dots, p_{n} \end{array}

can be given as :

\begin{array}{l} V a r (X) = σ_{x}^{2} = \sum_{i = 1}^{n} (x_{i} - μ)^{2} p_{i} \end{array}

\begin{array}{l} V a r (X) = \sum_{i = 1}^{n} (x_{i})^{2} p_{i} + \sum_{i = 1}^{n} μ^{2} p_{i} - \sum_{i = 1}^{n} 2 x_{i} μ p_{i} \end{array}

\begin{array}{l} V a r (X) = \sum_{i = 1}^{n} (x_{i})^{2} p_{i} + μ^{2} \sum_{i = 1}^{n} p_{i} - 2 μ \sum_{i = 1}^{n} x_{i} p_{i} \end{array}

Here,

\begin{array}{l} \sum_{i = 1}^{n} x_{i} p_{i} = μ \end{array}

(Mean of

\begin{array}{l} X \end{array}

) and

\begin{array}{l} \sum_{i = 1}^{n} p_{i} = 1 \end{array}

(sum of probabilities of all the outcomes of an event is 1). Substituting the values, we get

\begin{array}{l} V a r (X) = \sum_{i = 1}^{n} (x_{i})^{2} p_{i} + μ^{2} - 2 μ^{2} \end{array}

\begin{array}{l} σ_{x}^{2} = V a r (X) = \sum_{i = 1}^{n} (x_{i})^{2} p_{i} - μ^{2} \end{array}

\begin{array}{l} V a r (X) = E (X^{2}) - [E (X)]^{2} \end{array}

Where,

\begin{array}{l} E (X^{2}) = \sum_{i = 1}^{n} (x_{i})^{2} p_{i} \end{array}

and

\begin{array}{l} E (X) = \sum_{i = 1}^{n} x_{i} p_{i} \end{array}

An illustration of application of the concept is given below.

Example

Illustration 2: Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the mean and variance of the number of aces.

Solution: Let

\begin{array}{l} X \end{array}

be a random variable denoting the number of aces. Possible values of

\begin{array}{l} X \end{array}

are 0, 1,2.

\begin{array}{l} P (X = 0) = P (n o n - a c e a n d n o n - a c e) \end{array}

\begin{array}{l} P (n o n - a c e) \times P (n o n - a c e) \end{array}

\begin{array}{l} \frac{48}{52} \times \frac{48}{52} = \frac{144}{169} \end{array}

\begin{array}{l} P (X = 1) = P (a c e a n d n o n - a c e o r n o n - a c e a n d a c e) \end{array}

\begin{array}{l} P (a c e a n d n o n - a c e) + P (n o n - a c e a n d a c e) \end{array}

\begin{array}{l} P (a c e) \times P (n o n - a c e) + P (n o n - a c e) \times P (a c e) \end{array}

\begin{array}{l} \frac{4}{52} \times \frac{48}{52} + \frac{48}{52} \times \frac{4}{52} = \frac{24}{169} \end{array}

\begin{array}{l} P (X = 2) = P (a c e a n d a c e) \end{array}

\begin{array}{l} P (a c e) \times P (a c e) \end{array}

\begin{array}{l} \frac{4}{52} \times \frac{4}{52} = \frac{1}{169} \end{array}

Thus, the probability distribution can be given as,

X	0	1	2
P(X)	$\begin{array}{l} \frac{144}{169} \end{array}$	$\begin{array}{l} \frac{24}{169} \end{array}$	$\begin{array}{l} \frac{1}{169} \end{array}$

\begin{array}{l} E (X) = μ = \sum_{i = 1}^{n} x_{i} p_{i} = \frac{0.144}{169} + 1. \frac{24}{169} + 2. \frac{1}{169} \end{array}

\begin{array}{l} 0 + \frac{24}{169} + \frac{2}{169} = \frac{26}{169} \end{array}

\begin{array}{l} E (X^{2}) = \sum_{i = 1}^{n} (x_{i})^{2} p_{i} = 0^{2} . \frac{144}{169} + 1^{2} . \frac{24}{169} + 2^{2} . \frac{1}{169} \end{array}

\begin{array}{l} 0 + \frac{24}{169} + \frac{4}{169} = \frac{28}{169} \end{array}

\begin{array}{l} V a r (X) = E (X^{2}) - [E (X)]^{2} = \frac{28}{169} - (\frac{26}{169})^{2} = \frac{24}{169} \end{array}

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MATHS Related Links

Convex Polygon

Subsets

Magnitude Of A Vector

Types Of Triangles

Limits And Derivatives Class 11

Math Formula

Supplementary Angles

Mode Formula In Statistics

Circles Class 9

Area Of Shapes

Mean And Variance Of Random Variable