Mean of a Discrete Probability Distribution
Mean of a Discrete Probability Distribution
- It is the central value or the average of its corresponding probability mass function.
- It is often regarded as a measure of central location of a random variable
- It is the weighted average of the values that a random variable can take
(the weights are provided by the proability distribution) - It is sometimes called the expected value or expectation of
or - It is computed using the formula:
Variance of a Discrete Probability Distribution
Variance of a Discrete Probability Distribution
- It shows the variability or the scatterings of the random variables.
- It also shows the distance of a random variable from the mean.
- It is computed using the formula:
- Alternatively, it can be computed using the formula:
Standard Deviation of a Discrete Probability Distribution
- The standard deviation
can simply be computed by taking the square-root of the variance. - Its formula is given as one of these two forms:
Problems involving Mean and Variance
Problem
Find the mean for a random variable
defined as the number of tails in two tosses of a coin. Solution
Step 1: Create a Probability Distribution Table
First, we find the number of outcomes inb the sample space and the number of tails in each outcome.
Possible Outcome Number of Tails Then, we the frequency of each number of tails.
Since the total number of outcomes is, we divide the frequency by to get its probability.
Number of Tails Frequency Step 2: Next, we multiply
to its corresponding probability .
Number of Tails Frequency Step 3: Finally, we sum up each
to get the mean. Now, find the variance of the random variable
. Solution
Step 1: Find the mean
of the random variable.
From the previous problem, we found out that the mean. Step 2: Find
.
To do this, we simply use our probability distribution for, then subtract from each .
Number of Tails = 0 Step 3: Find
In this step, we simply square each result we got from the previous step.
Number of Tails Step 4: Multiply
and .
Number of Tails Step 5: Add each product from the previous step.