96. The probability of the engine starting with one attempt is constant and equal to p. Formulate the distribution series of the random variable $X$ - the number of engine starts per one attempt. Find the mathematical expectation and variance of $X$.
S o l u t i o n. The test consists of an attempt to start the engine. In the test, event $A$ - the engine starting - may or may not occur. The probability of event $A$ occurring in one test is $p$; the probability of the opposite event $\bar{A}$ - the engine will not start, is $q=1-p$. The random variable $X$ as a result of the test can take one of the following two possible values: $x_{1}=0$, $x_{2}=1$. The probabilities of these values are $q$ and $p$ respectively.

The random variable $X$ has the following distribution:

$$
X: \begin{array}{lll}
x_{i} & 0 & 1 \\
p_{i} & q & p
\end{array}
$$

The expected value and variance of the variable $X$ are:

$$
\begin{aligned}
& M(X)=0 \cdot q+1 \cdot p=p \\
& D(X)=M\left(X^{2}\right)-M^{2}(X)=p-p^{2}=p(1-p)=p q
\end{aligned}
$$