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5. Suppose Y has c.d.f. F(t) = 0, 1/6, 3/6, 1, for t < -1; for-1 < t < 2; for 2 ≤ t < 22; for t≥ 22. (a) Determine the p.m.f. or p.d.f. of Y, whichever is appropriate. (b) Recall that F(t) = P{Y ≤ t}. Define F(t-) by

5. Suppose Y has c.d.f. F(t) = 0, 1/6, 3/6, 1, for t < -1; for-1 < t < 2; for 2 ≤ t < 22; for t≥ 22. (a) Determine the p.m.f. or p.d.f. of Y, whichever is appropriate. (b) Recall that F(t) = P{Y ≤ t}. Define F(t-) by

A First Course in Probability (10th Edition)
A First Course in Probability (10th Edition)
10th Edition
ISBN:9780134753119
Author:Sheldon Ross
Publisher:Sheldon Ross
Chapter1: Combinatorial Analysis
Section: Chapter Questions
Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...
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### Problem 5: Distribution and Probability Analysis Suppose \( Y \) has a cumulative distribution function (c.d.f.) defined as: \[ F(t) = \begin{cases} 0, & \text{for } t < -1;\\ 1/6, & \text{for } -1 \leq t < 2;\\ 3/6, & \text{for } 2 \leq t < 22;\\ 1, & \text{for } t \geq 22. \end{cases} \] **(a)** Determine the probability mass function (p.m.f.) or probability density function (p.d.f.) of \( Y \), whichever is appropriate. **(b)** Recall that \( F(t) = \text{P}\{ Y \leq t \} \). Define \( F(t^-) \) by: \[ F(t^-) = \lim_{s \to t^-} F(s) \] - What is \( F(t^-) \) the probability of? **(c)** What is \( F(t) - F(t^-) \) the probability of? **(d)** For \( F \) as given in this problem, what is \( F(21) - F(21^-) \) and \( F(22) - F(22^-) \)? **(e)** Let \( I \) be the interval defined by \(\{ t \in \mathbb{R} \mid F(t) \geq 1/27 \}\). The interval \( I \) is of the form \( I = [b, \infty) \) where \( b \) equals what? **(f)** Let \( I \) be the interval defined by \(\{ t \in \mathbb{R} \mid F(t) \geq 2/6 \}\). The interval \( I \) is of the form \( I = [b, \infty) \) where \( b \) equals what?
Transcribed Image Text:### Problem 5: Distribution and Probability Analysis Suppose \( Y \) has a cumulative distribution function (c.d.f.) defined as: \[ F(t) = \begin{cases} 0, & \text{for } t < -1;\\ 1/6, & \text{for } -1 \leq t < 2;\\ 3/6, & \text{for } 2 \leq t < 22;\\ 1, & \text{for } t \geq 22. \end{cases} \] **(a)** Determine the probability mass function (p.m.f.) or probability density function (p.d.f.) of \( Y \), whichever is appropriate. **(b)** Recall that \( F(t) = \text{P}\{ Y \leq t \} \). Define \( F(t^-) \) by: \[ F(t^-) = \lim_{s \to t^-} F(s) \] - What is \( F(t^-) \) the probability of? **(c)** What is \( F(t) - F(t^-) \) the probability of? **(d)** For \( F \) as given in this problem, what is \( F(21) - F(21^-) \) and \( F(22) - F(22^-) \)? **(e)** Let \( I \) be the interval defined by \(\{ t \in \mathbb{R} \mid F(t) \geq 1/27 \}\). The interval \( I \) is of the form \( I = [b, \infty) \) where \( b \) equals what? **(f)** Let \( I \) be the interval defined by \(\{ t \in \mathbb{R} \mid F(t) \geq 2/6 \}\). The interval \( I \) is of the form \( I = [b, \infty) \) where \( b \) equals what?
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Step 1

From the given information,

The cdf of Y is,

Ft=0, for t<-116, for -1≤t<236, for 2≤t<221, for t≥22

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