Question

You invest in two startup companies focused on data science. Thanks to your growing expertise in this area, each company will reach unicorn status (valued at $1 billion) with probability 3/4, independent of the other company. Let random variable X be the total number of companies that reach unicorn status. X can take on the values 0, 1, and 2.
Give a complete expression for the probability mass function of X.
Give a complete expression for the cumulative probability function of X.
Compute E(X).
Compute var(X).

Answer 1

Answer:

a) $P(X=0)=(2C0)(0.75)^0 (1-0.75)^{2-0}=0.0625$  

$P(X=1)=(2C1)(0.75)^1 (1-0.75)^{2-1}=0.375$  

$P(X=2)=(2C2)(0.75)^2 (1-0.75)^{2-2}=0.5625$  

So then the probability mass function would be given by:

X        |    0                1               2

P(X)    | 0.0625     0.375      0.5625

b) $F(X) = 0.0625 , X=0$

$F(X)=0.0625+0.375 = 0.4375 , X \leq 1$

$F(X) = 0.4375+0.5625= 1 , X \leq 2$

c) The expected value for a binomail distribution is given by:

$E(X) = np = 2*0.75 = 1.5$

And the variance is given by:

$Var(X) = np(1-p) = 2*0.75*(1-0.75) = 0.375$

Step-by-step explanation:

Previous concepts

A Bernoulli trial is "a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted". And this experiment is a particular case of the binomial experiment.

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

The probability mass function for the Binomial distribution is given as:  

$P(X)=(nCx)(p)^x (1-p)^{n-x}$  

Where (nCx) means combinatory and it's given by this formula:  

$nCx=\frac{n!}{(n-x)! x!}$  

Solution to the problem

Let X the random variable of interest "total number of companies that reach inicorn status", and we can assume that the distribution for X is binomial with possible values X =0,1,2:  

$X \sim Binom(n=2, p=3/4 = 0.75)$  

Part a

Using the probability mass function we can find the probabilities for all the possible values of X like this:

$P(X=0)=(2C0)(0.75)^0 (1-0.75)^{2-0}=0.0625$  

$P(X=1)=(2C1)(0.75)^1 (1-0.75)^{2-1}=0.375$  

$P(X=2)=(2C2)(0.75)^2 (1-0.75)^{2-2}=0.5625$  

So then the probability mass function would be given by:

X        |    0                1               2

P(X)    | 0.0625     0.375      0.5625

Part b

For this case the cumulative distribution function would be given by:

$F(X) = 0.0625 , X=0$

$F(X)=0.0625+0.375 = 0.4375 , X \leq 1$

$F(X) = 0.4375+0.5625= 1 , X \leq 2$

Part c

The expected value for a binomail distribution is given by:

$E(X) = np = 2*0.75 = 1.5$

And the variance is given by:

$Var(X) = np(1-p) = 2*0.75*(1-0.75) = 0.375$