3. Suppose X is a continuous-type random variable with p.d.f. f(t). How would you compute each of the following? (a) P{X ≤ 3.7}. (b) P{X ≤t}. (Make sure that you do not use the symbol t to mean two different things in your answer.) (c) E[X]. (d) E[X²]. (e) E[e¹x].

3. Suppose X is a continuous-type random variable with p.d.f. f(t). How would you compute each of the following? (a) P{X ≤ 3.7}. (b) P{X ≤t}. (Make sure that you do not use the symbol t to mean two different things in your answer.) (c) E[X]. (d) E[X²]. (e) E[e¹x].

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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Question

3. Suppose X is a continuous-type random variable with p.d.f. f(t). How would
you compute each of the following?
(a) P{X ≤ 3.7}.
(b) P{X ≤t}. (Make sure that you do not use the symbol t to mean two
different things in your answer.)
(c) E[X].
(d) E[X²].
(e) E[e¹x].

Expert Solution

Step 1

Given that the pdf of a continuous random variable $X$ is $f (t)$ . That is $X ~ f (t)$ for $t \in D$ , where D is the domain where the function $f (t)$ is defined. Then, for an arbitrary $a$ , the probability $P (X \leq a) = \int_{- \infty}^{a} f (t) d t$ , and the expectation is defined as $E [X] = \int_{- \infty}^{\infty} t f (t) d t$ .

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