Conditional probability

Question

I am new to Mathematica. I am asking how to define a conditinal probability in Mathematica.

Here is an example of what I want to do:

Assume that $P(x_b\,|\,x_i)=\left\{ \begin{array}{} 1-p & : x_b=0\\ p & : x_b=x_i \end{array} \right.$

where $P(x_b|x_i)$ is the conditional probability which takes 0 with probability of $1-p$ and takes the value of $x_i$ with probability $p$, where $0\le p\le 1$.

Also, we have $x_i\in U[3,5]$, by which I mean $x_i$ is uniformly distributed between 3 and 5.

I know that: $P(x_i,x_b)=P(x_b|x_i)P(x_i)$

I know how to define $P(x_i)$ in Mathematica using

xi=UniformDestribution[{3,5}]

but I want to find how to define conditional probability $P(x_b|x_i)$ in Mathematica. I tried to use the following:

xb = TransformedDistribution[xi xb, xb \[Distributed] BernoulliDistribution[p]]

and then multiply $x_b$ with $x_i$, but when I run

RandomVariate[xb xi]

I get error message

The valid numeric parameters of distribution TransformedDistribution[[FormalX]UniformDistribution[{3,5}], [FormalX][Distributed]BernoulliDistribution[0.1]] are expected. Use DistributionParameterAssumptions to obtain the parameter assumptions.

Is there any way to find $P(x_i,x_b)$ which equals $P(x_b|x_i)P(x_i)$

Answer 1

The joint density function of $x_b$ and $x_i$ can be written as

f[xb_, xi_] := Piecewise[{{(1 - p)/2, xb == 0 && 3 <= xi <= 5}, {p/2, 3 <= xb <= 5 && 3 <= xi <= 5}}]

• Added May 12, 2015 *

While I haven't been able to get Mathematica to display the joint density function from the TransformedDistribution function, that function does supply some of the appropriate characteristics of the joint distribution function:

xbxi = TransformedDistribution[{\[Alpha] xi, 
   xi}, {xi \[Distributed] 
    UniformDistribution[{3, 5}], \[Alpha] \[Distributed] 
    BernoulliDistribution[p]}]
Mean[xbxi]
Covariance[xbxi]
RandomVariate[xbxi /. p -> 1/3, 10]