Let Y be a binomial random variable with n = 1 and success

Step 1 of 2

Let Y be a Binomial random variable with n = 1, p be the probability of success, and q be the probability of failure

The claim is to find the probability and distribution function for Y.

The probability for Y is given by

\(\mathrm{P}(\mathrm{Y}=\mathrm{K})=\left({ }_{k}^{n}\right) p^{k}(1-p)^{n-k}\)

we have n = 1

Then, \(\mathrm{P}(\mathrm{Y}=\mathrm{K})=p^{k}(1-p)^{1-k}\)

If k = 0, P(Y = K) = 1 - p

If k = 1, P(Y = K) =  p

The probability function of Y is

\(P(Y=K)=\left\{\begin{array}{cc} 1-p, & \text { if } k=0 \\ p, & k=1 \\ 0, & \text { otherwise } \end{array}\right.\)

Hence, the distribution function is

\(\mathrm{F}(\mathrm{y}=) P(Y \leq y)=\left\{\begin{array}{c} 0, \text { if } y<0 \\ 1-p, \text { if } 0 \leq y<1 \\ 1, \text { if } y \geq 1 \end{array}\right.\)