- Discrete random variables can only have a finite or countably infinite number of values. The number of heads in 10 tosses of a fair coin, the toss number of the first head if a fair coin is tossed until a head appears, or the number of green balls selected in the example given above.
- Continuous random variables can assume any of an uncountably infinite set of values. For example, if a point is picked at random on the interval from [0,1], there are an uncountably infinite number of values that could be picked.
| t | 0 | 1 | 2 | 3 |
| P[T=t] | 1/8 | 3/8 | 3/8 | 1/8 |
Notice that the sum of the probabilities is 1. This is true for any discrete random variable.
Run the experiment of tossing a fair coin 3 times 1000 times updating after every 100 tosses. To do this link here, and when the page opens click the red die in front of number 4. Set the number of coins at 3. After running the experiment 1000 times, what can you say about the theoretical (in blue) and actual (in red) probability distributions of the number of heads?
In all examples of discrete random variables, the probabilities in the probability distribution table give the 'long-term' proportion of times that the random variable assumes each possible value.
For examples 3 and 4, you can use this link to a simulation of the situation. When the page opens click the red die in front of number 4 to open the simulation. When the simulation opens, set N to 20, set R to 10, the number of red balls, and set n, the sample size to 2. Select with or without replacement as appropriate for the example. The blue graph and the text below it will show probabilities for each number of red balls.
Consider again the count of heads in 3 tosses of a fair coin. If this experiment is repeated, say 10 times, and the number of heads in each series of 3 tosses is counted, you will have a set of numbers like 0,1,3,1,2,2,1,1,3,0. The average of these numbers is the average value for the random variable, the number of heads in 3 tosses of a fair coin. To see what happens in a larger number of runs of the experiment, again link here, and when the page opens click the red die in front of number 4. Set the number of coins at 3 and run the experiment 100 times. What is the average number of heads per 3 tosses? Now reset and run the experiment 1000 times. What is the average number of heads per 3 tosses?
The long-term average number of heads is called the expected value of the random variable, the number of heads in 3 tosses of a fair coin. This expected value can be found for most random variables. Think of expected value as the average value of a random variable.
There is an easier way to find the expected value of this (or any) discrete random variable. If the experiment of tossing the coin 3 times is repeated for a large number, N, times, the experiment will end in 0 heads n0 times, in 1 head n1 times, in 2 heads n2 times, and in 3 heads n3 times. The total number of heads is 0 n0 + 1 n1 + 2 n2 + 3 n3, and the average number of heads per run of the experiment is
(0 n0 + 1 n1 + 2 n2 + 3 n3)/N = 0 (n0/N) + 1 (n1/N) + 2 (n2/N) + 3 (n3/N)
For large N, (n0/N) ~ P[0 Heads], (n1/N) ~ P[1 Head], (n2/N) ~ P[2 Heads], (n3/N) ~ P[3 Heads], so the average number of heads per run of the experiment is
0 P[0 Heads] + 1P[1 Head] + 2 P[2 Heads] + 3 P[3 Heads]
This is called the Expected Value or Mean and is denoted, for a general random variable X, by E[X]. It can be computed by
Using this formula on the random variable T, the total number of heads in 3 tosses of a fair coin, you get
µ=E[T] = 0 (1/8) + 1 (3/8) + 2 (3/8) + 3 (1/8) = 12/8 = 3/2 = 1.5. This can be interpreted as the average number of heads per sequence of 3 tosses if the experiment is repeated a large number of times.
Just as you are able to find the average value for a random variable, so you can also find the standard deviation of the random variable. In the case of a random variable, the standard deviation is given by
For random variable T, the total number of heads in 3 tosses of a fair coin, the standard deviation computed by the rightmost formula is
SD[T] = Square Root of [02 (1/8) + 12 (3/8) + 22 (3/8) + 32 (1/8) - (3/2)2] = Square Root of [3/4] = 31/2/2
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\begin{align} &\overbrace{\color{#f44}{\sum_{k = 0}^{n}{n \choose k}\pars{1 \over 2}^{k} \pars{1 \over 2}^{n - k}}} ^{\substack{\ds{First\ Round:}\\[1mm] \ds{\color{#f44}{k}\ \mbox{heads}}}}\,\,\, \overbrace{\color{#44f}{\sum_{j = 0}^{n - k}{n - k \choose j}\pars{1 \over 2}^{j} \pars{1 \over 2}^{n - k - j}}} ^{\substack{\ds{Second\ Round:} \\[1mm] \ds{\color{#44f}{j}\ \mbox{heads}}}} \\[2mm] &\ \times\pars{k + j} \\[5mm] = &\ {1 \over 2^{2n}}\sum_{k = 0}^{n}\sum_{j = 0}^{n - k}{n \choose k} {n - k \choose j}2^{k}\pars{k + j} = \bbx{{3 \over 4}\,n} \\[5mm] &\ \stackrel{\ds{n\ =\ 10}}{\ds{\implies}}\quad\bbx{7.5} \\ & \end{align}Union Public Service Commission (UPSC) has released the NDA Result II 2022 (Name Wise List) for the exam that was held on 4th September 2022. Earlier, the roll number wise list was released by the board. A total number of 400 vacancies will be filled for the UPSC NDA II 2022 exam. The selection process for the exam includes a Written Exam and SSB Interview. Candidates who get successful selection under UPSC NDA II will get a salary range between Rs. 15,600 to Rs. 39,100.