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Citrus2011 [14]
3 years ago
8

The lifetime of two light bulbs are modeled as independent and exponential random variables X and Y, with parameters lambda and

mu, respectively. The time at which one of those two light bulb first burns out is Z = min (X, Y), what is the PDF of Z?
Mathematics
1 answer:
Scorpion4ik [409]3 years ago
7 0

Answer:

\bf h(x)=max(\lambda,\mu) e^{-max(\lambda,\mu) x}\;(x\geq0)

Step-by-step explanation:

The PDF of X is

\bf f(x)=\lambda e^{-\lambda x}\;(x\geq0)

The PDF of Y is

\bf g(x)=\mu e^{-\mu x}\;(x\geq0)

The means of X and Y are respectively,

\bf \displaystyle\frac{1}{\lambda}\;,\displaystyle\frac{1}{\mu}

so we can see that the larger the parameter, the smaller the mean. Hence the PDF of Z = min(X, Y) is an exponential with the largest parameter of the two.

Therefore, the PDF of Z is

\bf h(x)=max(\lambda,\mu) e^{-max(\lambda,\mu) x}\;(x\geq0)

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Step-by-step explanation:

\frac{2x - 1}{4}  +  \frac{x}{3}  = 2

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\frac{6x - 3}{12}  +  \frac{4x}{12}  =  \frac{24}{12}

\frac{6x - 3 + 4x}{12}  =  \frac{24}{12}

\frac{10x - 3}{12}  =  \frac{24}{12}

multiply both sides by 12

12 \times  \frac{10x - 3}{12}  =  \frac{24}{12}  \times 12

12s cancel out

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add three to both sides

10x - 3 + 3 = 24 + 3

10x = 27

divide both sides by 10

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The tens on the left side cancel

x =  \frac{27}{10}  = 2.7

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2 years ago
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maksim [4K]

Answer:

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Step-by-step explanation:

First, we convert the mixed numbers into improper fractions: 1\frac{3}{7} =\frac{10}{7}, and 2\frac{3}{4}=\frac{11}{4}. Now, we multiply the fraction using the multiplication rule, \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}: \frac{10}{7} \times \frac{11}{4} = \frac{110}{28} = \frac{55}{14}.

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