Suppose that the random variables Y1 and Y2 have joint probability distribution function. f(y1, y2) = 2, 0 ≤ y1 ≤ 1,
Suppose that the random variables Y1 and Y2 have joint probability distribution function. f(y1, y2) = 2, 0 ≤ y1 ≤ 1,
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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Question
Suppose that the random variables
probability distribution function .
Y1
and
Y2
have joint f(y1, y2) =
| 2, | 0 ≤ y1 ≤ 1, 0 ≤ y2 ≤ 1, 0 ≤ y1 + y2 ≤ 1, |
| 0, | elsewhere |
(a)
Use R to calculate
P(Y1 ≥ 1⁄6 | Y2 ≤ 1⁄5).
(Round your answer to four decimal places.)P(Y1 ≥ 1⁄6 | Y2 ≤ 1⁄5) =
(b)
Use R to calculate
P(Y1 ≥ 1⁄6 | Y2 = 1⁄5).
(Round your answer to four decimal places.)P(Y1 ≥ 1⁄6 | Y2 = 1⁄5) =
![**Title: Understanding Joint Probability Distribution Functions**
**Description:**
Suppose that the random variables \( Y_1 \) and \( Y_2 \) have a joint probability distribution function defined as:
\[ f(y_1, y_2) = \begin{cases}
2, & 0 \leq y_1 \leq 1, \ 0 \leq y_2 \leq 1, \ 0 \leq y_1 + y_2 \leq 1, \\
0, & \text{elsewhere}.
\end{cases} \]
**Tasks:**
(a) **Use R to Calculate Conditional Probability**
Calculate \( P(Y_1 \geq 1/6 \mid Y_2 \leq 1/5) \). Provide your answer rounded to four decimal places.
\[ P(Y_1 \geq 1/6 \mid Y_2 \leq 1/5) = \] [Text box for answer input] [Incorrect symbol]
(b) **Use R to Calculate Conditional Probability (Alternate Scenario)**
Calculate \( P(Y_1 \geq 1/6 \mid Y_2 = 1/5) \). Provide your answer rounded to four decimal places.
\[ P(Y_1 \geq 1/6 \mid Y_2 = 1/5) = \] [Text box for answer input] [Incorrect symbol]
**Explanation:**
This exercise involves evaluating the conditional probabilities in a defined region of a joint probability distribution function. The function \( f(y_1, y_2) \) is defined over a triangular region where \( 0 \leq y_1 + y_2 \leq 1 \). Use R programming to compute these probabilities with precision for further understanding.
**Note:**
The scenarios presented in (a) and (b) involve different conditions for \( Y_2 \), showcasing how such alterations impact probability calculations.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe9b346ae-ef24-4d99-babf-e95763a360cd%2F305b23b4-2e74-4d28-a351-137fbbe30ee6%2Fe11wt3m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Joint Probability Distribution Functions**
**Description:**
Suppose that the random variables \( Y_1 \) and \( Y_2 \) have a joint probability distribution function defined as:
\[ f(y_1, y_2) = \begin{cases}
2, & 0 \leq y_1 \leq 1, \ 0 \leq y_2 \leq 1, \ 0 \leq y_1 + y_2 \leq 1, \\
0, & \text{elsewhere}.
\end{cases} \]
**Tasks:**
(a) **Use R to Calculate Conditional Probability**
Calculate \( P(Y_1 \geq 1/6 \mid Y_2 \leq 1/5) \). Provide your answer rounded to four decimal places.
\[ P(Y_1 \geq 1/6 \mid Y_2 \leq 1/5) = \] [Text box for answer input] [Incorrect symbol]
(b) **Use R to Calculate Conditional Probability (Alternate Scenario)**
Calculate \( P(Y_1 \geq 1/6 \mid Y_2 = 1/5) \). Provide your answer rounded to four decimal places.
\[ P(Y_1 \geq 1/6 \mid Y_2 = 1/5) = \] [Text box for answer input] [Incorrect symbol]
**Explanation:**
This exercise involves evaluating the conditional probabilities in a defined region of a joint probability distribution function. The function \( f(y_1, y_2) \) is defined over a triangular region where \( 0 \leq y_1 + y_2 \leq 1 \). Use R programming to compute these probabilities with precision for further understanding.
**Note:**
The scenarios presented in (a) and (b) involve different conditions for \( Y_2 \), showcasing how such alterations impact probability calculations.
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