Consider a plane with a maximum capacity of 50 passengers. Suppose it is known that on a particular the discrete random variable X (which counts the number of passengers who actually show up for the flight) has the following pmf: 45 46 47 48 49 0.05 0.1 0.12 0.14 0.25 a) With what probability will all passengers who show up for the flight have a seat on this flight? b) What is the expected number of passengers who show up for this flight? What is the associated variance? x P(X=x) 50 51 52 53 54 55 0.17 0.06 0.05 0.03 0.02 0.01
Consider a plane with a maximum capacity of 50 passengers. Suppose it is known that on a particular the discrete random variable X (which counts the number of passengers who actually show up for the flight) has the following pmf: 45 46 47 48 49 0.05 0.1 0.12 0.14 0.25 a) With what probability will all passengers who show up for the flight have a seat on this flight? b) What is the expected number of passengers who show up for this flight? What is the associated variance? x P(X=x) 50 51 52 53 54 55 0.17 0.06 0.05 0.03 0.02 0.01
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![**Probability and Expected Value Analysis for Flight Capacity**
Consider a plane with a maximum capacity of 50 passengers. Suppose it is known that on a particular flight, the discrete random variable \( X \) (which counts the number of passengers who actually show up for the flight) has the following probability mass function (pmf):
\[
\begin{array}{c|ccccccccccc}
x & 45 & 46 & 47 & 48 & 49 & 50 & 51 & 52 & 53 & 54 & 55 \\
\hline
P(X = x) & 0.05 & 0.1 & 0.12 & 0.14 & 0.25 & 0.17 & 0.06 & 0.05 & 0.03 & 0.02 & 0.01 \\
\end{array}
\]
### Questions
**a) With what probability will all passengers who show up for the flight have a seat on this flight?**
To find the probability that all passengers who show up have a seat, we need to sum the probabilities for \( X \leq 50 \).
\[
P(X \leq 50) = P(X = 45) + P(X = 46) + P(X = 47) + P(X = 48) + P(X = 49) + P(X = 50)
\]
\[
= 0.05 + 0.1 + 0.12 + 0.14 + 0.25 + 0.17 = 0.83
\]
So, the probability that all passengers who show up for the flight have a seat is \( 0.83 \) or 83%.
**b) What is the expected number of passengers who show up for this flight? What is the associated variance?**
1. **Expected Number \( E(X) \)**
The expected number of passengers can be calculated using the definition of the expected value for discrete random variables:
\[
E(X) = \sum_{x} x \cdot P(X = x)
\]
\[
= 45 \cdot 0.05 + 46 \cdot 0.1 + 47 \cdot 0.12 + 48 \cdot 0.14 + 49 \cdot 0.25 + 50 \cd](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb19fb600-654f-4bb5-880f-15449a731741%2F259d5694-cdfc-4b83-ba4f-0d1fa1ca4379%2Ffk25x1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Probability and Expected Value Analysis for Flight Capacity**
Consider a plane with a maximum capacity of 50 passengers. Suppose it is known that on a particular flight, the discrete random variable \( X \) (which counts the number of passengers who actually show up for the flight) has the following probability mass function (pmf):
\[
\begin{array}{c|ccccccccccc}
x & 45 & 46 & 47 & 48 & 49 & 50 & 51 & 52 & 53 & 54 & 55 \\
\hline
P(X = x) & 0.05 & 0.1 & 0.12 & 0.14 & 0.25 & 0.17 & 0.06 & 0.05 & 0.03 & 0.02 & 0.01 \\
\end{array}
\]
### Questions
**a) With what probability will all passengers who show up for the flight have a seat on this flight?**
To find the probability that all passengers who show up have a seat, we need to sum the probabilities for \( X \leq 50 \).
\[
P(X \leq 50) = P(X = 45) + P(X = 46) + P(X = 47) + P(X = 48) + P(X = 49) + P(X = 50)
\]
\[
= 0.05 + 0.1 + 0.12 + 0.14 + 0.25 + 0.17 = 0.83
\]
So, the probability that all passengers who show up for the flight have a seat is \( 0.83 \) or 83%.
**b) What is the expected number of passengers who show up for this flight? What is the associated variance?**
1. **Expected Number \( E(X) \)**
The expected number of passengers can be calculated using the definition of the expected value for discrete random variables:
\[
E(X) = \sum_{x} x \cdot P(X = x)
\]
\[
= 45 \cdot 0.05 + 46 \cdot 0.1 + 47 \cdot 0.12 + 48 \cdot 0.14 + 49 \cdot 0.25 + 50 \cd
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