State which of the following is not the probability distribution of a random variable. Give reasons for your answer.
$Z$ $3$ $2$ $1$ $0$ $-1$ $P(Z)$ $0.3$ $0.2$ $0.4$ $0.1$ $0.05$
| $Z$ | $3$ | $2$ | $1$ | $0$ | $-1$ |
| $P(Z)$ | $0.3$ | $0.2$ | $0.4$ | $0.1$ | $0.05$ |
(N/A) For a probability distribution of a random variable,two conditions must be satisfied:
$1$. Each probability $P(Z_i)$ must be $\ge 0$.
$2$. The sum of all probabilities $\sum P(Z_i)$ must be equal to $1$.
In the given table,the sum of probabilities is:
$\sum P(Z) = 0.3 + 0.2 + 0.4 + 0.1 + 0.05 = 1.05$.
Since the sum of probabilities is $1.05$,which is not equal to $1$,the given table does not represent a probability distribution of a random variable.
$1$. Each probability $P(Z_i)$ must be $\ge 0$.
$2$. The sum of all probabilities $\sum P(Z_i)$ must be equal to $1$.
In the given table,the sum of probabilities is:
$\sum P(Z) = 0.3 + 0.2 + 0.4 + 0.1 + 0.05 = 1.05$.
Since the sum of probabilities is $1.05$,which is not equal to $1$,the given table does not represent a probability distribution of a random variable.
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