New Cable Subseribers. Spectrum provides cable television...

New Cable Subseribers. Spectrum provides cable television and Internet service to millions of customers. Suppose that the management of Spectrum subjectively assesses a probability distribution for the number of new subscribers next year in the state of New York as follows. $$ \begin{array}{cc} \boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) \\ 100,000 & 10 \\ 200,000 & 20 \\ 300,000 & 25 \\ 400,000 & .30 \\ 500,000 & .10 \\ 600,000 & .05 \end{array} $$ a. Is this probability distribution valid? Explain. b. What is the probability Spectrum will obtain more than 400,000 new subscribers? c. What is the probability Spectrum will obtain fewer than 200,000 new subscribers?

New Cable Subseribers. Spectrum provides cable television and Internet service to millions of customers. Suppose that the management of Spectrum subjectively assesses a probability distribution for the number of new subscribers next year in the state of New York as follows.
$$
\begin{array}{cc}
\boldsymbol{x} & \boldsymbol{f}(\boldsymbol{x}) \\
100,000 & 10 \\
200,000 & 20 \\
300,000 & 25 \\
400,000 & .30 \\
500,000 & .10 \\
600,000 & .05
\end{array}
$$

a. Is this probability distribution valid? Explain.
b. What is the probability Spectrum will obtain more than 400,000 new subscribers?
c. What is the probability Spectrum will obtain fewer than 200,000 new subscribers?

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Key Concepts

Discrete Random Variable

A discrete random variable is one that takes on a countable number of distinct values. In statistics and probability, this concept is important for modeling outcomes where each possible value can be associated with a specific probability.

Probability Distribution Validation

Validating a probability distribution for a discrete random variable requires checking that each individual probability is between 0 and 1 and that the sum of these probabilities is exactly 1. This ensures that the distribution represents a complete and consistent set of outcomes for the random variable.

Event Probability Computation

Computing the probability of an event involves adding the probabilities of all individual outcomes that satisfy the event’s conditions. This concept is key when determining the likelihood of ranges or conditions being met within the distribution.

Tail Probability

Tail probability refers to the probability that a discrete random variable will take on a value either above or below a certain threshold. It is determined by summing the probabilities of outcomes that fall in the tail regions of the distribution.

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