Pls help check my answer for this probability question.

In summary, the lifespan of a species of plant is a random variable T (tens of days) with a probability density function of f(t) = (1/8)e^(-t/8) for t > 0 and 0 otherwise. The cumulative distribution function of T is F(t) = 0 for t < or = 0 and 1 - e^(-t/8) for t > 0. Using this, the probability that a randomly chosen plant of this species has a lifespan of more than 20 days is 0.779. The expected lifespan of this species is 80 days.
  • #1
denian
641
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the lifespan of a species of plant is a random variable T ( tens of days ). the probability density function is given by

f(t) = { (1/8)e^(-t/8) , t > 0
{ 0, otherwise

(i) find the cumulative distribution function of T and sketch its graph

i get this answer

F(t) = { 0, t < or = 0
{ 1 - e^(-t/8) , t > 0


(ii) find the probability, to three decimal places, that a plant of that species randomly chosen has a lifespan of more than 20 days

i do it this way..

F(2) = ...
= 0.221

P(lifespan > 20 days) = 1 - 0.221 = 0.779


(iii) calculate the expected lifespan of that species of plant
i get E(t) = 8
therefore, expected lifespan of that species is 80 days.


please tell me if i am correct in the questions above. sorry, i tried to use Latex but my attemp failed.
 
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  • #2


Your answers are correct! Good job! As for using LaTeX, it may take some practice to get used to the syntax and formatting. Keep trying and you'll get the hang of it.
 
  • #3


Your answers for parts (i) and (ii) are correct. The cumulative distribution function is the integral of the probability density function, and you have correctly integrated to get the expression for F(t). Your calculation for the probability that a plant has a lifespan of more than 20 days is also correct.

For part (iii), your answer for the expected lifespan is incorrect. The expected value of a random variable is calculated by integrating the variable multiplied by the probability density function. In this case, it would be:

E(t) = ∫t*f(t) dt = ∫t*(1/8)e^(-t/8) dt

Using integration by parts, you should get E(t) = 8. This means that the expected lifespan of this species of plant is 8 tens of days, or 80 days.

Overall, your answers are correct. Keep in mind that for future probability questions, it may be helpful to show your work or how you arrived at your answers, especially if you are using a different method than the one provided in the solution. This will help to ensure that your understanding of the concept is correct. Good job!
 

1. What is the definition of probability?

Probability is the measure of the likelihood that an event will occur. It is typically expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.

2. How do you calculate probability?

Probability can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For example, if you are rolling a die and want to find the probability of rolling a 4, you would divide 1 (the number of favorable outcomes) by 6 (the total number of possible outcomes) to get a probability of 1/6 or approximately 0.167.

3. What is the difference between theoretical and experimental probability?

Theoretical probability is based on mathematical calculations and assumes that all outcomes are equally likely. Experimental probability, on the other hand, is based on actual data collected from experiments or observations. It can vary from theoretical probability if the sample size is not large enough.

4. How do you interpret probability?

Probability can be interpreted as a measure of how likely or unlikely an event is to occur. For example, a probability of 0.75 can be interpreted as there being a 75% chance of the event happening.

5. What are some real-life applications of probability?

Probability is used in many real-life situations, such as predicting the weather, assessing risk in insurance and finance, and making decisions in sports and gambling. It is also used in various scientific fields, such as genetics, epidemiology, and physics.

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