Procedure for the experiments in the following section

Part 1

CDF

Click on the button to generate a random number "c".

Find the inverse image of the generated random number "c" for the given random variable $X$ and select the appropriate inverse image.

Based on the inverse image calculated and the probabiliy measure given, find the CDF of the random variable $X$ at "c" and enter the value in the input box.

Click on the "Submit" button to check your answer.

Click on the "Reset" button to reset the experiment.

CDF Properties

In this experiment, you will be shown 4 graphs of the cumulative distribution function (CDF) of 4 different random variables.You have to verify if the given plots satisfy all the properties of a CDF.

To do this, you have to verify each graph individually.

First click on a gaph to select it. The selected graph number will be showed below.

After this, select the Property of CDF that you think the graph does not satisfy. If it satisfies all the properties, select the last option.

After selecting the option, the observation will be displayed.

CDF to PDF/PMF

Click on the button to randomly generate a plot of CDF of a valid RV. Note that the only types of RV used here are either continuous Uniform RV of a discrete RV with 5 non-zero point.

Identify the type of RV from the plot and select from the dropdown.

If you choose continuous RV, enter the PDF value and the range of values of RV in which the PDF is non-zero. For example, if the plot is a straight line from (1,0) to (2,0), the PDF value here is 1, the leftmost non-zero point is 1 and the rightmost non-zero point is 2.

If you choose discrete RV, enter the PMF values in increasing order of value of RV.

Click on the "Submit" button to check your answer.

Click on the "Reset" button to reset the experiment.

Part 2

Bernoulli RV

Here, we are considering a coin toss experiment as a Bernoulli random variable $X$. Head is mapped to $X=1$ and Tail is mapped to $X=0$.

Choose a value for the probability of getting a Head on tossing a coin, denoted by $p$.

Enter the value of $p$ in the input box and click on the "Set" button.

Click on the "TOSS" button to simulate the coin toss experiment.

Observe the value of the Bernoulli random variable $X$ for each toss.

Click on the "Reset" button to reset the experiment.

Binomial RV

In this experiment, we are modelling the number of Heads we get when we flip a biased coin $n$ times as a Binomial RV. The probability of getting a head is given by $P(H) = p$.

Here, we are fixing the number of coin flips to $n = 10$.

You can set the value of $P(H)$ using the input box and then click on the "Set" button.

After setting the value of $P(H)$, you can click on the "TOSS" button to flip the coin once.

You can also click on the "Complete remaining toss" button to flip the coin remaining number of times at once.

After the coin is flipped 10 times, check observation to see how the value of the binomial random variable changes.

Repeat this experiment multiple times to see how the value of the binomial random variable changes. You can repeat the experiment by clicking on the "Reset" button.

Geoemetric RV

Choose a value for the probability of getting a Head on a coin toss, denoted by $p$.

Enter the value of $p$ in the input box and click on the "Set" button.

Click on the "TOSS" button to perform the coin toss experiment.

Keep on clicking the "TOSS" button to perform multiple coin tosses until you get the first Head.

Observe the value of the Geometric random variable $X$ for the experiment.

Repeat the experiment multiple times to observe the distribution of the Geometric random variable $X$.

Click on the "Reset" button to reset the experiment.

Poisson RV

In this experiment, we are trying to show that as $n \to \infty$ and $p \to 0$ such that $np$ is constant, the binomial random variable $X$ converges to a Poisson random variable with parameter $\lambda = np$.

Ypu have to conduct three experiments with different $n$ and $p$ values such that $np$ is constant.

A value for $\lambda$ is randomly generated. You have to set the $p$ and $n$ values for the three experiments such that $np = \lambda$.

To observe the convergence, you have to enter a value for $n$ between 10 to 50 for the first experiment, 50 to 100 for the second experiment, and 100 to 150 for the third experiment.

After entering these values, you can click on the "Set" button to set the values.

You can then click on the "Toss for 1 experiment at a time" button to conduct the experiments one by one or click on the "Complete all the 3 experiments at once" button to conduct all the experiments at once.

After the experiments are conducted, you can observe the number of heads and tails obtained in each experiment, the observations, and the graph showing the PDF of the three experiments and the Poisson distribution.