Spring_2022_Homework_4

Homework 4 Spring 2022 Due May 2, 2022 @ 12:00pm PST Name: Student ID: Section: Instructions Please read the following instructions carefully: 1. Please show all notation for probability statements. 2. Box your final answers. 3. Please verify that your scans are legible. 4. Please assign pages the the questions when submitting to gradescope. 5. This assignment is due via gradescope on the due date. Quy Nguyen 11372355 B

Homework 4 1. The height of male gira ↵ es follows a normal distribution with μ = 18 feet and σ = 0 . 9 feet 2 . (a) What is the probability that a male gira ↵ e has a height between 19 and 22 feet? Write out the solution in integral form and use the following R code to compute the numerical answer: pnorm (22 , 18 , 0 . 9) - pnorm (19 , 18 , 0 . 9). (b) What height would a male gira ↵ e need to be in order to be in the top 10 percent? Write out the equation we would need to solve in order to answer this question, then use the qnorm () function in R to find this score. (c) What is the probability that a male gira ↵ e will grow shorter than 16 feet. (d) The tallest male gira ↵ e in the world is measured to be 21 feet according to the Guinness World Records. What is the probability that a gira ↵ e would grow higher than 21 feet. Page 2 FIX )= ¥ E , e- G- a) 4202 f ? 0.91 ¥ e- ( × -18 ) " " " DX → Using R : pnorm (22,18/0.9) - pnorm (19,1%0.9) = 0.1332559 ? [ o÷ ☒ e -1 × -185/1-62 dx = 0.1 qnorm ( I -0.1 , 18,0 - 9) = 19.1534ft [ solve for n f " o.ae#Ee- 1 × -18141.62 d × 0 pnorm (16,18/0.9) = 0.013134 [ o.ae#ge-( × -18141.62 d × 21 1- pnorm / 21,18 , 0.9 ) = 0.0004291

Homework 4 3. Let X be a continuous random variable. Let f ( x ) = c ( x - 1) 3 and S X = [1 , 3]. Hint: ( x - 1) 3 = x 3 + 3 x - 3 x 2 - 1 (a) What value of c will make f ( x ) a valid density? (b) What is P ( X = 2)? (c) Find E ( X ). (d) What is P (1 < X < 2)? Page 4 / ? c ( x - 1) 3 dx = I [ 4C = 1 c = IT C / ? 43+3 × -3 × 2 - 1) dx = I c. [ ¥ + 3 ¥ - ✗ 3- × ] ? = I c [ ⾨ + ¥ - 27 - 3) _ ( 4- + 3- - l - l ) ) = I P ( ✗ = 2) = 0 Because ✗ is a continuous random variable ECX ) =/ ? 4- × ( x - 1) 3 dx = ? 44+3 × 2 - 3 × 3 - × ) dx = f- [ ¥ + ✗ 3- 3 ¥ - ¥ ] ? = ± , [ ( 2 ¥ + 27 - 2443--9-2 ) - ( § + I _ 3-4-12 ) ) = 2.6 f f ÷ , ( x - 1) 3 dx = tf ? ( ✗ 3+3 × -3 × 2 - 1) dx = 4- [ ¥ + 3 ¥ - ✗ 3- × ] ? = ÷ [ ( ¥ + ¥ - 8- 2) - ( 1-1+3-2 - I - 1) ] = IT [ IT ] = % = 0.0625

Homework 4 4. The battery life of a fully charge iPhone 13 Pro on a standby follows a normally distributed with mean μ = 7 hour and variance σ 2 = 1 hours. (a) Write out the integral form of the probability that a fully-charged iPhone 13 Pro last between 7 and 9 hours on standby. then use the pnorm function in R to find this probability. (b) What is the probability that a fully-charged iPhone 13 Pro last less than 5 hours? (c) What is the probability that a fully-charged iPhone 13 Pro last more than 9 hours? (d) What is the minimum time a fully-charged iPhone 13 Pro need to last in order to be in the top 10% best quality? Page 5 ☒ = ¥¥ et × -Ñ / 202 R : pnorm / 9,7 , 1) - pnorm (7/7,1)=0.4772499 § ¥ , e- ( × - 7) 42 µ 02=1 hr → 0 = 1hr2 5 P( ✗ < 5) = go ¥ , e- ( × - 7) 212 dx R :p norm ( 5,7 , 1) = 0.02275 PCX > 9) =/ of ¥ , e- 1 × -7142 d × R : I - pnorm (9/7,1) = 0.02275 R : qnorm (1-0.1/7,1) = 8.2816 hours

Homework 4 6. Ashley, who is a student at UCI commutes to school 5 days a week via Uber. Her wait time from requesting a ride until she is picked up follows a uniform distribution with a lower bound of 2 minutes and an upper bound of 10 minutes. (a) What is the probability that Ashley wait time from requesting a ride until she is picked up is more than 7 minutes? (b) What is the Ashley’s expected wait time? (c) What is the probability that Ashley wait time is between 3 and 5 minutes? Now suppose Mike who is another student from UCI also commutes to school 5 days a week via Uber. For question (d) and (e), assume that the wait time of each person is independent and identically distributed. (iid). (d) What is the probability that exactly one out of the two student’s wait time is less than 5 minutes. (e) What is the probability that at least one of students wait time is less than 5 minutes. Page 7 [ 2,10 ] P( × > 7) = I - P( ✗ ≤ 7) = I - ?¥ z = I - % P( × > 7) = 0.375 Check R : I - pun if (7,2/10) = 0.375 E- ( X ) = b ¥ = 10 ¥ = 6 minutes f- ( x ) = ¥ a = ¥ 2 = ¥ f) b- dx = ¥ / § = & - 3- = % = 0.25 Check R : punif ( 5 , 2,10 ) - pun if (3/2,10)=0.25 PIX , < 5) ◦ P( ✗ z > 5) R : pun if (5,2/10) . ( I - pun if (5,2/10) ) = 0.234375 I P( × , < 5) • P( ✗ z > 5) ← part (d) → 0.234375 a 2 PIX , < 5) . P( Xz < 5) / Add the 2 probabilities pun if (5,2/10) - punif (5,2/10) = 0.140625 < ⊥ 0.375

Homework 4 7. A new restaurant open up in downtown LA. The waiting time for a customer to be seated can be modeled as an exponential distribution. The averaged waiting time is 45 minutes. (a) What is the parameter of the Exponential distribution? What is it equal to in this problem? (b) What is the probability of a customer waits 30 to 40 minutes to be seated? (c) A customer has been waiting for 45 minutes but still haven’t been seated. What is the probability that this customer is seated after waiting for 60 minute since he lined in the queue? Page 8 F- Is ↑ T parameter 1 customer per 45 minutes 1=(401-1--130) = ( I - e- 40145 ) - ( I - e- 30145 ) = 0.1023 Check R : pexp (40,1/45) - pexp (30,1/45) = 0.1023 1- p( ✗ < 60 ) = I - I - e- 60145 ) = 0.2636 Check R : I - pexp (60,1/45) = 0.2636