CONTINUOUS PROBABILITY DISTRIBUTIONS CHAPTER 15 Area = 𝟏 𝟐 𝒃×𝒉= 𝟏 𝟐 ×𝟏×𝟐=𝟏 𝝆 𝒙 ≥𝟎 𝐟𝐨𝐫 𝐚𝐥𝐥 𝒙 𝐢𝐧 𝐭𝐡𝐞 𝐝𝐨𝐦𝐚𝐢𝐧 ∴ 𝝆 𝒙 𝒊𝒔 𝒂 𝒑𝒅𝒇

𝐥𝐢𝐦 𝒌→∞ 𝟏 𝒌 𝒇 𝒙 𝒅𝒙=𝟏

< 𝒂𝒏𝒅 ≤ > 𝒂𝒏𝒅 ≥ or between: For a continuous random variable, there is no difference between the inequality signs: < 𝒂𝒏𝒅 ≤ or between: > 𝒂𝒏𝒅 ≥

For any continuous random variable X, Pr(X=a) = 0

EXAMPLE 1

EXAMPLE 2

The longterm average, or Expected value, of the random variable is: MEASURES OF CENTRE FOR A CONTINUOUS PROBABILITY DISTRIBUTION The mode is the value of the random variable for which the highest value of f(x) occurs. The median is the value of the random variable for which there is an area of 0.5 to the left and 0.5 to the right. The median is also called the 50th percentile The longterm average, or Expected value, of the random variable is: 𝝁=𝑬 𝑿 = −∞ ∞ 𝒙𝒇 𝒙 𝒅𝒙

EXAMPLE 3:

EXAMPLE 4 A teacher travels by car to school and the journey time, t hours, has a probability density function given by: 𝑓 𝑡 = 10𝑐 𝑡 2 9𝑐(1−𝑡) 0 0≤𝑡≤0.6 0.6<𝑡≤1 otherwise where c is a constant. Find the value of c.    Sketch the graph of 𝑓 𝑡 .  Calculate the median travel time. d. Determine the probability that the travel time will be : i. More than 48 minutes.  ii. Between 24 minutes and 48 minutes.

MEASURES OF SPREAD FOR A CONTINUOUS RANDOM VARIABLE Variance and Standard Deviation Var(X) = 𝑬 𝑿 𝟐 − 𝝁 𝟐 and 𝛔= 𝐯𝐚𝐫(𝐗) 𝐰𝐡𝐞𝐫𝐞: 𝑬( 𝑿 𝟐 )= −∞ ∞ 𝒙 𝟐 𝒇 𝒙 𝒅𝒙

THE INTERQUARTILE RANGE (IQR) 𝐼𝑄𝑅=𝑞 −𝑝 where q = 75th Percentile and p = 25th percentile. 𝟎 𝒑 𝒇 𝒙 𝒅𝒙=𝟎.𝟐𝟓 𝟎 𝒒 𝒇 𝒙 𝒅𝒙=𝟎.𝟕𝟓

𝒇 𝒕 ={ 𝒎𝒕(𝟏𝟎𝟎− 𝒕 𝟐 ) 𝟎 𝟎<𝒕≤𝟏𝟎 𝐞𝐥𝐬𝐞𝐰𝐡𝐞𝐫𝐞 EXAMPLE 5 The time t, in minutes that Jamie spends riding his bike to work is a continuous random variable with the probability density function: 𝒇 𝒕 ={ 𝒎𝒕(𝟏𝟎𝟎− 𝒕 𝟐 ) 𝟎 𝟎<𝒕≤𝟏𝟎 𝐞𝐥𝐬𝐞𝐰𝐡𝐞𝐫𝐞 Calculate the value of m. Calculate the exact mean time that Jamie takes to ride to work. Calculate the standard deviation, correct to two decimal places, of the time it takes Jamie to get to work on his bike. Calculate the interval within which lie the middle 50% of times it takes for Jamie to get to work, correct to two decimal places.