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 and check your answer.

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.

Select a graph by clicking on it first and then, select the Property of CDF that you think the graph does not satisfy. If it satisfies all the properties, select the last option.

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.

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

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 $p$.

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

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

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

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.

Observe the value of the Geometric random variable $X$ for 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 $\lambda = np$.

To observe the convergence, you have to enter a value for $n$ and a value of $\lambda$

You can see the two histograms matching and the error values decreasing.