Let Y be normal standard distribution and Z be Poisson distribution with i.e lambda = 1. Suppose a random variable such as X is defined as follows: (flip a coin)
X = Y (if Heads occurs)
X = Z (if Tails occurs)
I want to know the figure of the CDF of the X?!
I tried to achieve this which I do know whether it is true or not:
$F_X(t) = P(X \le t) = P(X \le t | heads)P(heads) + P(X \le t | tails)P(tails)=$
$=\frac{1}{2}P(Y \le t) + \frac{1}{2} P(Z \le t) = \frac{1}{2}[F_Y(t) + F_Z(t)]$
so, how can I plot the figure of the CDF of X?
Best Answer
Here is a plot made in Mathematica. The blue curve is the standard normal CDF, the orange piecewise function is the Poisson CDF, and the green is the desired mixture. Due to the discrete nature of the Poisson distribution, its CDF has jump discontinuities at each nonnegative integer.