Answer:
See explanation
Step-by-step explanation:
Solution:-
The random variable, Y be the temperature of chemical reaction in degree fahrenheit be a linear expression of a random variable X : The temperature in at which a certain chemical reaction takes place.
Y = 1.8*X + 32
- The median of the random variate "X" is given to be equal to "η". We can mathematically express it as:
P ( X ≤ η ) = 0.5
- Then the median of "Y" distribution can be expressed with the help of the relation given:
P ( Y ≤ 1.8*η + 32 )
- The left hand side of the inequality can be replaced by the linear relation:
P ( 1.8*X + 32 ≤ 1.8*η + 32 )
P ( 1.8*X ≤ 1.8*η ) ..... Cancel "1.8" on both sides.
P ( X ≤ η ) = 0.5 ...... Proven
Hence,
- Through conjecture we proved that: (1.8*η + 32) has to be the median of distribution "Y".
b)
- Recall that the definition of proportion (p) of distribution that lie within the 90th percentile. It can be mathematically expressed as the probability of random variate "X" at 90th percentile :
P ( X ≤ p_.9 ) = 0.9 ..... 90th percentile
- Now use the conjecture given as a linear expression random variate "Y",
P ( Y ≤ 1.8*p_0.9 + 32 ) = P ( 1.8*X + 32 ≤ 1.8*p_0.9 + 32 )
= P ( 1.8*X ≤ 1.8*p_0.9 )
= P ( X ≤ p_0.9 )
= 0.9
- So from conjecture we saw that the 90th percentile of "X" distribution is also the 90th percentile of "Y" distribution.
c)
- The more general relation between two random variate "Y" and "X" is given:
Y = aX + b
Where, a : is either a positive or negative constant.
- Denote, (np) as the 100th percentile of the X distribution, so the corresponding 100th percentile of the Y distribution would be : (a*np + b).
- When a is positive,
P ( Y ≤ a*p_% + b ) = P ( a*X + b ≤ a*p_% + b )
= P ( a*X ≤ a*p_% )
= P ( X ≤ p_% )
= np_%
- When a is negative,
P ( Y ≤ a*p_% + b ) = P ( a*X + b ≤ a*p_% + b )
= P ( a*X ≤ a*p_% )
= P ( X ≥ p_% )
= 1 - np_%
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