The amount of milk (in litres) in a shop at the beginning of any day is a random amount 𝑋X from which a random amount 𝑌Y (in litres) is sold during that day. Assume that the joint density function of 𝑋X and 𝑌Y is given by𝑓𝑋𝑌(𝑥,𝑦)={1500≤𝑥≤10,0≤𝑦≤𝑥0otherwisef XY (x,y)={ 501 0 0≤x≤10,0≤y≤xotherwise .Find the probability that amount of milk left at the end of day is less than 4 litres. Write your answer correct to two decimal points.
Question
The amount of milk (in litres) in a shop at the beginning of any day is a random amount 𝑋X from which a random amount 𝑌Y (in litres) is sold during that day. Assume that the joint density function of 𝑋X and 𝑌Y is given by𝑓𝑋𝑌(𝑥,𝑦)={1500≤𝑥≤10,0≤𝑦≤𝑥0otherwisef XY (x,y)={ 501 0 0≤x≤10,0≤y≤xotherwise .Find the probability that amount of milk left at the end of day is less than 4 litres. Write your answer correct to two decimal points.
Solution
To find the probability that the amount of milk left at the end of the day is less than 4 litres, we need to consider two cases:
- The amount of milk at the beginning of the day is less than 4 litres.
- The amount of milk at the beginning of the day is more than 4 litres, but the amount sold is such that the remaining amount is less than 4 litres.
For the first case, the probability is the integral of the joint density function from 0 to 4 for x and from 0 to x for y.
For the second case, the probability is the integral of the joint density function from 4 to 10 for x and from x-4 to x for y.
Let's calculate these probabilities:
-
∫ from 0 to 4 [ ∫ from 0 to x (1/50) dy ] dx = ∫ from 0 to 4 [ (1/50)*x ] dx = [ (1/100)*x^2 ] from 0 to 4 = 16/100 = 0.16
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∫ from 4 to 10 [ ∫ from x-4 to x (1/50) dy ] dx = ∫ from 4 to 10 [ (1/50)*4 ] dx = [ (4/50)*x ] from 4 to 10 = (4/50)*6 = 24/100 = 0.24
Adding these probabilities gives us the total probability that the amount of milk left at the end of the day is less than 4 litres:
0.16 + 0.24 = 0.40
So, the probability that the amount of milk left at the end of the day is less than 4 litres is 0.40 or 40%, correct to two decimal points.
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