Question

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7946268-dt-content-rid-23953695 1/courses/EGD126 1913TP1/MZB126 S1 2019 P3.pd 4/4 Part C (Based off week 9 workshop content) 1. Smokey, the rescue cat, now lives a life of luxury and peace in his new furever-home with his owner Sam. When Sam arrives home on time, he finds his cat asleep 8/10 days. However, when he comes home late, his cat is awake 7/10 days. Answer the following questions given Sam arrives home on time from work 40% of the time: a State the events. b State the provided probabilities (at least 3, however you can include more if you wish to include compliments) c If Smokey is asleep, what is the probability that Sam is late? 2. When late, the time (in minutes) it takes for Sam to get home follows the pdf given below f(t) ne-/o0. a what value of η makes this a valid pdf? b What is the expected number of minutes Sain is late by, given le is late? c What is the probability that Sam is running more than 30 mins late, given he is late?

Answer 1

1.a) Sam arrives home on time

Sam arrives home late

When Sam arrives home on time , Smokey is asleep

When Sam arrives home late , Smokey is asleep

b) P(Sam arrives home on time ) = 0.40

P(Smokey is asleep I Sam arrives home late ) = 7/10

P(Smokey is asleep I Sam arrives home on time  ) = 8/10

c) P(Sam arrives home late I Smokey is asleep)

= P(Smokey is asleep I Sam arrives home late ).P(Sam arrives home late) / P(Smokey is asleep)

  P(Smokey is asleep) = P(Smokey is asleep I Sam arrives home late ).P(Sam arrives home late)

+P(Smokey is asleep I Sam arrives home on time  ). P(Sam arrives home on time )

= 7/10 *0.60 + 8/10*0.40

= 0.74

Therefore ,

P(Sam arrives home late I Smokey is asleep) = 0.7*0.6 / 0.74 = 0.57

2 a)

For valid pdf

$\int_{0}^{\infty }f(t)dt =1$

$\Rightarrow \int_{0}^{\infty }\eta e^{-t/30}dt =1$

Solving for integral

$\eta = 1/30$

b) The above pdf is pdf for exponential distribution with parameter $\eta$

Expectation is $1/\eta$

Therefore , expected number of minutes Sam is late by is 30 minutes.

c)

$P(t>30)=\int_{30}^{\infty }(1/30)e^{-t/30}dt$

= $-e^{-\infty }-(-e^{-1})$

= 0+0.3679

= 0.3679