Answer :
Question:
Consider the following exponential probability density function.
f(x) = 1/5e^(−x/5) for x ≥ 0
(a) Write the formula for P(x ≤ x0). (b) Find P(x ≤ 4). (Round your answer to four decimal places.) (c) Find P(x ≥ 5). (Round your answer to four decimal places.) (d) Find P(x ≤ 6). (Round your answer to four decimal places.) (e) Find P(4 ≤ x ≤ 6). (Round your answer to four decimal places.)
Answer:
(a) P(x ≤ x0) = 1 - e^(−x0/5)
(b) P(x ≤ 4) = 0.5506
(c) P(x ≥ 5) = 0.3678
(d) P(x ≤ 6) = 0.6988
(e) P(4 ≤ x ≤ 6) = 0.1482
Step-by-step explanation:
The standard form of the exponential probability density function is given by
f(x) = 1/μe^(−x/μ)
Where μ is the mean, for the given problem μ = 5
(a) Write the formula for P(x ≤ x0)
P(x ≤ x0) = 1 - e^(−x0/5)
(b) Find P(x ≤ 4)
P(x ≤ 4) = 1 - e^(−4/5)
P(x ≤ 4) = 1 - 0.4493
P(x ≤ 4) = 0.5506
(c) Find P(x ≥ 5)
P(x ≥ 5) = e^(−5/5)
P(x ≥ 5) = 0.3678
(d) Find P(x ≤ 6)
P(x ≤ 6) = 1 - e^(−6/5)
P(x ≤ 6) = 1 - 0.3011
P(x ≤ 6) = 0.6988
(e) Find P(4 ≤ x ≤ 6)
P(4 ≤ x ≤ 6) = e^(−4/5) - e^(−6/5)
P(4 ≤ x ≤ 6) = 0.4493 - 0.3011
P(4 ≤ x ≤ 6) = 0.1482
(a): The required value is [tex]P\left ( x\leq x_{0} \right )=1-e^{-x_{0/5}}[/tex]
(b): The required value is [tex]P(x\leq 4)=0.5507[/tex]
(c): The required value is[tex]\\P(x\geq 5)=0.3679[/tex]
(d): The required value is[tex]P(x\le 6)=0.6988[/tex]
(e): The required value is[tex]P(4 \leq x \leq6)= 0.1481[/tex]
Probability density function:
The probability density function (PDF) is used to define the random variable’s probability coming within a distinct range of values, as opposed to taking on anyone's value.
Given function is,
[tex]f\left ( x \right )=1/5e^{-x/5}for \ x%u 22650\geq 0[/tex]
Part(a): Write the formula for [tex]P\left ( x\leq x_{0} \right )[/tex]
[tex]P\left ( x\leq x_{0} \right )=1-e^{-x_{0/5}}[/tex]
Part(b): Find [tex]P(x \le 4)[/tex],
[tex]P(x\geq 4)\\P(x\leq 4)=1-e^{-4/5} \\P(x\leq 4)=0.5507[/tex]
Part(c): Find [tex]P(x\geq 5)[/tex]
[tex]\\P(x\geq 5)=1-P}(x\leq 5)\\\\P(x\leq 5)= 1 - e^{-5/5}\\\\P(x \leq 5) = 0.6321P(x \geq 5)\\ = 1 - 0.6321P(x \geq 5) \\= 0.3679[/tex]
Part(d): Find [tex]P(x \leq 6)[/tex]
[tex]P(x\le 6)=F(6)\\=1-e^{-6/5}\\=0.6988[/tex]
Part(e): Find [tex]P(4\leq x\leq 6)[/tex]
[tex]P(4 \leq x \leq6) = P(x \leq 6) - P(x \leq 4)P(4 \leq x \leq 6) = e^{-6/5} - e^{-4/5}\\P(4 \leq x \leq 6) = 0.4493-0.3012\\P(4 \leq x \leq 6) = 0.1481[/tex]
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