Problem 14 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

The probability distribution of the random variable \(X\) is shown in the accompanying table: $$\begin{array}{lcccccc} \hline x & -5 & -3 & -2 & 0 & 2 & 3 \\ \hline P(X=x) & .17 & .13 & .33 & .16 & 11 & .10 \\ \hline\end{array}$$ Find a. \(P(X \leq 0)\) b. \(P(X \leq-3)\) c. \(P(-2 \leq X \leq 2)\)

Step-by-Step Solution

Answer

a. \(P(X \leq 0)=0.79\) b. \(P(X \leq -3)=0.30\) c. \(P(-2 \leq X \leq 2)=0.60\)

Step 1: a. Calculating \(P(X \leq 0)\)

To calculate the probability of X being less than or equal to 0, we need to add the probabilities of X being -5, -3, -2, and 0. From the probability distribution, we have: \(P(X=-5)=0.17\), \(P(X=-3)=0.13\), \(P(X=-2)=0.33\), and \(P(X=0)=0.16\) Therefore, \(P(X \leq 0)=P(X=-5)+P(X=-3)+P(X=-2)+P(X=0)=0.17+0.13+0.33+0.16=0.79\)

Step 2: b. Calculating \(P(X \leq-3)\)

To calculate the probability of X being less than or equal to -3, we need to add the probabilities of X being -5 and -3. From the probability distribution, we have: \(P(X=-5)=0.17\) and \(P(X=-3)=0.13\) Therefore, \(P(X \leq -3)=P(X=-5)+P(X=-3)=0.17+0.13=0.30\)

Step 3: c. Calculating \(P(-2 \leq X \leq 2)\)

To calculate the probability of X being between -2 and 2 (inclusive), we need to add the probabilities of X being -2, 0, and 2. From the probability distribution, we have: \(P(X=-2)=0.33\), \(P(X=0)=0.16\), and \(P(X=2)=0.11\) (Note: There is a typo in the question for P(X=2), it should be 0.11 rather than 11.) Therefore, \(P(-2 \leq X \leq 2)=P(X=-2)+P(X=0)+P(X=2)=0.33+0.16+0.11=0.60\) The results are as follows: a. \(P(X \leq 0)=0.79\) b. \(P(X \leq -3)=0.30\) c. \(P(-2 \leq X \leq 2)=0.60\)

Key Concepts

Random Variable

A random variable is a numerical quantity that assigns a measurable value to each outcome of a probabilistic event or experiment. For instance, when throwing a dice, the outcome is a random variable that can take values from 1 to 6 inclusive. In our exercise, the random variable is denoted by the letter \(X\) and refers to different possible scores, such as -5, -3, -2, 0, 2, and 3.

These scores represent specific outcomes from a certain context, perhaps a game or a series of events. The corresponding probability of each outcome occurring is listed alongside it, creating what's known as a probability distribution. It's essential to grasp this concept because understanding what a random variable represents in a problem helps clarify the nature of the probability we're trying to calculate. Each value of the random variable has an associated probability, which gives us insight into how likely that outcome is.

Cumulative Probability

Cumulative probability is a term in statistics that refers to the probability that a random variable takes on a value less than or equal to a particular number. Think of it as accumulating probabilities of all outcomes up to and including the one you're interested in.

For example, in the exercise \(P(X \leq 0)\) represents the cumulative probability for the random variable \(X\) for all values up to 0. To find it, you simply add up the probabilities for all possible values of the random variable that are less than or equal to 0. This sum evidently results in the cumulative probability, which quantitatively gauges the likelihood of the random variable falling within that specified range. Understanding cumulative probability is crucial when you want to assess risk, anticipate outcomes, or make decisions based on all available information up to a certain point.

Probability Calculation

Probability calculation involves determining the likelihood of an occurrence using mathematical reasoning. In the realm of random variables, this often entails summing the probabilities of individual outcomes, particularly when these outcomes are discrete and distinct.

It's not just about pure addition, though. Accurate probability calculation requires careful consideration of all possible outcomes, vigilant adherence to the rules of probability, and sometimes involves using formulas, such as those for cumulative probabilities or expected values.

Missteps to Avoid

• Assuming probabilities for non-mutually exclusive events can simply be added without correcting for any overlap.
• Forgetting to ensure that all probabilities in a distribution must add up to 1, providing a check on the calculation's correctness.
• Overlooking potential typographical errors, as seen in our exercise where '11' should have been '0.11' for \(P(X=2)\), which can lead to errors in calculation.

By keeping these pitfalls in mind and following a step-by-step approach as shown in our solutions, you can accurately determine the probability of various events associated with a random variable.