Question

a The length of time (in years) until a particular radioactive particle decays is a random variablet with probability density function defined by f(t) = 8 e - 8t fort in [0, 0o). Find the mean of the distribution, the standard deviation of the distribution, and the probability that the random variable is between the mean and 1 standard deviation above the mean. b The length (in centimeters) of the leaf of a certain plant is a continuous random variable with probability density function defined by f(x) = ï for x in [1, 3.2). Find the mean of the distribution, the standard deviatio the distribution, and the probability that the random variable is between the mean and 1 standard deviation above the mean. C Find a z-score such that 20% of the total area is to the left of z. (Hint: Use the table backwards.) Click here to view page 1 of the table. Click here to view page 2 of the table. Choose the closest z-score such that 20% of the total area is to the left of z.

please answer all 3

Answer 1

ANSWER:
0 - ey even that :- f (t)=8 est fort in ay Now; Let {1+)= 8.2-8t te [0,0) tse it at EL?).- Polana au e da -st dt té P MP 8 12 82 8 Head = 1 / 3 Now, 6(72 = 8 ftest dt 0 8x13 83 2 / 32 JVCT) - ET2-LET32 2 / 2 82 :: glandard deviation - VVLT) 1

P( Between mean and 9 standard deveation above the mean) - PLOCT($) 78 . est sie at 0 [-e -engt 7/8 O ine § 1-036 7819 0.6321 Ang b) Given that - F(x) = 5 Forex in [2.3.2] Probability density function f(x) = I fore x en (1302) Now, se een a f(x) da 3.2 M = fx. ut de ។ 3.2 5 튜 fda 1 .3.2 1 [x2] [43.2)2-(02] 5 22

11 1094-1) 게 902y M 22 21 s 10 2 -1 M pe Variance o² = ( 2² f(x) dx ee2 302 22통d 20192 1 302 &모da - Youl ) 14 90 야 (J- SS [Car)-³-11 h

@ - 4.81 333 -4.41 0.403335 02 = 0.40833 o 0.6350826 standard deviation (0.) = '006351 & cooby 0.64 is f 5 da 11 2.1 5 Il [22].0.64 2.1 na [10.6432–1212 22 5 22 (0.4096- 4.4) 5_(-4.2004) $10.909 10.909 Ang

6 C) PLZZZ) = 20 0/0 P( 222) 50020 P[z (-0.8416] - 020 Z .0.84.16 2= -0.8 - 0 • ៩ ។ (pounded) standard .: using Nooma! table]