Expectation (Mean) and Variance of a Random Variable
Suppose we have a probability experiment we are performing, and we have defined some random variable(R.V.) according to our needs( like we did in some previous examples). Now, each time experiment is performed, our R.V. takes on a different value. But we want to know that if we keep on doing the experiment a thousand times or an infinite number of times, what will be the average value of the random variable?
Expectation
The mean, expected value, or expectation of a random variable X is written as E(X) or [Tex]\mu_{\textbf{X}}. [/Tex] If we observe N random values of X, then the mean of the N values will be approximately equal to E(X) for large N.
For a random variable X which takes on values x1, x2, x3 … xn with probabilities p1, p2, p3 … pn. Expectation of X is defined as,
[Tex]\sum_{i=1}^{N} x_{i}p_{i} [/Tex]
i.e it is weighted average of all values which X can take, weighted by the probability of each value.
To see it more intuitively, let’s take a look at this graph below,
Now in the above figure, we can see both the Random Variables have the almost same ‘mean’, but does that mean that they are equal? No. To fully describe the properties/behavior of a random variable, we need something more, right?
We need to look at the dispersion of the probability distribution, one of them is concentrated, but the other is very spread out near a single value. So we need a metric to measure the dispersion in the graph.
Variance
In Statistics, we have studied that the variance is a measure of the spread or scatter in the data. Likewise, the variability or spread in the values of a random variable may be measured by variance.
For a random variable X which takes on values x1, x2, x3 … xn with probabilities p1, p2, p3 … pn and the expectation is E[X]
The variance of X or Var(X) is denoted by, [Tex]E[X – u]^{2} = \sum (x_{i}-\mu)^{2}p_{x_{i}} = E[X^{2}] – (E[X])^{2} [/Tex]
Let’s calculate the mean and variance of a random variable probability distribution through an example:
Example: Find the variance and mean of the number obtained on a throw of an unbiased die.
Answer:
We know that the sample space of this experiment is {1,2,3,4,5,6}
Let’s define our random variable X, which represents the number obtained on a throw.
So, the probabilities of the values which our random variable can take are,
P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = [Tex]\frac{1}{6} [/Tex]
Therefore, the probability distribution of the random variable is,
X 1 2 3 4 5 6 Probabilities [Tex]\frac{1}{6} [/Tex] [Tex]\frac{1}{6 } [/Tex] [Tex]\frac{1}{6} [/Tex] [Tex]\frac{1}{6} [/Tex] [Tex]\frac{1}{6} [/Tex] [Tex]\frac{1}{6} [/Tex] E[X] = [Tex]\sum p_{x_{i}}x_{i} \\ \hspace{0.9cm} = 1 \times \frac{1}{6} + 2 \times \frac{1}{6} + 3 \times \frac{1}{6} + 4 \times \frac{1}{6} + 5 \times \frac{1}{6} + 6 \times \frac{1}{6} \\ \hspace{0.9cm} = \frac{21}{6} [/Tex]
Also, E[X2] = [Tex]1^{2} \times \frac{1}{6} + 2^{2}\times\frac{1}{6} + 3^{2}\times\frac{1}{6} + 4^{2}\times\frac{1}{6} + 5^{2}\times\frac{1}{6} + 6^{2}\times\frac{1}{6} \\ \hspace{0.9cm} = \frac{91}{6} \\ [/Tex]
Thus, Var(X) = E[X2] – (E[X])2
= [Tex](\frac{91}{6}) – (\frac{21}{6})^{2} = \frac{91}{6} – \frac{441}{36} = \frac{35}{12} [/Tex]
So, therefore mean is [Tex]\frac{21}{6} [/Tex]and variance is [Tex]\frac{35}{12} [/Tex]
Probability Distribution – Function, Formula, Table
A probability distribution is an idealized frequency distribution. In statistics, a frequency distribution represents the number of occurrences of different outcomes in a dataset. It shows how often each different value appears within a dataset.
Probability distribution represents an abstract representation of the frequency distribution. While a frequency distribution pertains to a particular sample or dataset, detailing how often each potential value of a variable appears within it, the occurrence of each value in the sample is dictated by its probability.
A probability distribution, not only shows the frequencies of different outcomes but also assigns probabilities to each outcome. These probabilities indicate the likelihood of each outcome occurring.
In this article, we will learn what is probability distribution, types of probability distribution, probability distribution function, and formulas.
Table of Content
- What is Probability Distribution?
- Probability Distribution Definition
- Random Variables
- Random Variable Definition
- Types of Random Variables in Probability Distribution
- Probability Distribution of a Random Variable
- Probability Distribution Formulas
- Expectation (Mean) and Variance of a Random Variable
- Expectation
- Variance
- Different Types of Probability Distributions
- Discrete Probability Distributions
- Bernoulli Trials and Binomial Distributions
- Binomial Distribution
- Cumulative Probability Distribution
- Probability Distribution Function
- Probability Distribution Table
- Prior Probability
- Posterior Probability
- Solved Questions on Probability Distribution