3. Continuous Probability Distributions

Transcription

1 3.1 Continuous probability distributions 3. Continuous Probability Distributions K The normal probability distribution A continuous random variable X is said to have a normal distribution if it has a probability density function 1 p(x) = 2πσ 2 exp( (x µ)2 2σ 2 ) - < x <. The normal distribution is defined by two parameters, the mean value of X, µ and the variance of X, σ 2. Note: To indicate that a random variable has a normal distribution with mean µ and variance σ 2, we often write X ~ N(µ, σ 2 ). For continuous probability distributions, probabilities are defined by areas under the probability density curve. For example, if X ~ N(µ, σ 2 ), then Pr(a X b) = a b p(x) dx = 1 b 2πσ 2 a exp( (x µ)2 2σ 2 ) dx, which is equal to the area under the probability density curve between x = a and x = b. With a graphics calculator, this can be determined directly by using the normalcdf( command. The ShadeNorm( command will also display the probability as an area under the probability density curve. K.1 Determining normal probabilities Task : If X~N(10, 2 2 ), determine: 1. Pr(7 X 11) 2. Pr(X 8) 3. Pr(X 13.5) The normalcdf( command saves having to use tables to calculate normal probabilities. The normalcdf( command is located in the DISTR(ibution) menu under DISTR ( 2nd [DISTR] ( VARS )).

2 3.2 Continuous probability distributions Step 1. Evaluate Pr(7 X 11) Start on the HOME screen (press 2nd [QUIT] if you are not on the HOME screen), and CLEAR. (1) Go to the DISTR menu ( 2nd [DISTR] ( VARS )). Use the arrow key to move down the DISTR menu to option 2: normalcdf( and select by pressing ENTER. This pastes the normalcdf( command onto the HOME screen. (2) Complete the command as follows: normalcdf(7,11,10,2) lower value of X upper value of X µ Note: On the TI-83 Plus we use the standard deviation σ =2 (here) rather than the variance σ 2.! (3) Finally, press ENTER to evaluate. Thus, Pr(7 X 11)= (correct to 4 decimal places).

3 3.3 Continuous probability distributions Step 2. Evaluate Pr(X 8) To evaluate Pr(X 8) we need to recognise that we are implying that the lower limit for X is -. Thus, we rewrite Pr(X 8) as Pr(- <X 8) and proceed as before. (1) Starting on the HOME screen, call up the normalcdf( command and complete as follows: normalcdf( - E99, 8, 10, 2) Note: On the graphics calculator - is replaced by a very large negative number E99 = E is obtained by pressing 2nd [EE]. (2) Press ENTER to evaluate. Thus, Pr(X 8)= (correct to 4 decimal places). Step 3. Evaluate Pr(X 13.5) To evaluate Pr(X 13.5 ) we need to recognise that we are implying that the upper limit for X is so that we rewrite Pr(X 13.5) as Pr( 13.5 X< ) and proceed as before. (1) Starting on the HOME screen, call up the normalcdf( command and complete as follows: normalcdf( 13.5, E99,10, 2) Note: On the graphics calculator is replaced by a very large number, E99 = E is obtained by pressing 2nd [EE]. (2) Finally, press ENTER to evaluate. Thus, Pr(X 13.5)= Pr(13.5 X< )= (correct to 4 decimal places). Exercises 1. If Z~N(0, 1 2 ), show that, correct to 4 decimal places: (1) Pr( 1.5 Z 2) = (2) Pr(Z 0.8) = (3) Pr(Z 1.6)= (4) Pr( 1 Z 1)= If X~N(100, 20), show that, correct to 4 decimal places:

4 3.4 Continuous probability distributions (1) Pr(88 X 112) = (2) Pr(100 X 105) = (3) Pr(X 107) = (4) Pr(X 97) = K.2 Graphically displaying normal probabilities Tasks: 1. If X~N(10, 2 2 ), determine Pr(7 X 11) and display the normal curve with the appropriate area shaded. 2. If X~N(100, 10 2 ), determine Pr(- <X 105) and display the normal curve with the appropriate area shaded. The ShadeNorm( command determines normal probabilities by finding the appropriate area under the normal curve. It also draws the normal curve and shades in the area. The ShadeNorm( command is located in the DISTR(ibution)menu ( 2nd [DISTR] ( VARS )) under DRAW. Step 1. If X~N(10, 2 2 ), determine and display Pr(7 X 11). Start on the HOME screen (press 2nd [QUIT] if you are not on the HOME screen), CLEAR. (1) Go to the DISTR menu ( 2nd [DISTR] ( VARS ) ) and use the arrow key to move across to the DRAW submenu. Option 1: ShadeNorm( will be automatically highlighted. Press ENTER and the ShadeNorm( command will be pasted onto the HOME screen. (2) Complete the command as follows: ShadeNorm(7,11,10,2) lower value of x upper value of x µ!

5 3.5 Continuous probability distributions (3) As we are going to draw a graph, we will need to set the viewing window first. To do this, press WINDOW and enter the following settings: Xmin = 2 (set X min =µ 4σ) Xmax = 18 (set X max =µ+4σ) Xscale= 1 (choose an appropriate value) Ymin = 0.05 (use a small negative value to show the x-axis ) Ymax = 0.25 (try 0.5 σ to start) Yscale= 0.05 (choose an appropriate value) Xres=1 (4) As a precaution, to avoid other graphs being plotted at the same time, go to (i) Y= and either CLEAR or deselect all functions. (ii) press 2nd [STATPLOT] and turn all plots OFF. (5) Return to the HOME screen ( 2nd [QUIT]) and press ENTER to complete the task. Thus, Pr(7 X 11) = area under curve = (correct to 4 decimal places). Note: Before proceeding to the next activity press 2nd [DRAW] ( PRGM ) ENTER to clear the drawing from the screen.

6 3.6 Continuous probability distributions 2. If X~N(100, 10 2 ), determine and display Pr(X 105). To evaluate Pr(X 105) we need to recognise that we are implying that the lower limit for X is -. Thus we rewrite Pr(X 105) as Pr(- <X 105) and proceed as before. (1) Starting on the HOME screen, call up the ShadeNorm( command and complete as follows: ShadeNorm( - E99, 105, 100, 10) Note: On the graphics calculator - is replaced by a very large number, - E99 = E is obtained by pressing 2nd [EE]. (2) As we are going to draw a graph, we will need to set the viewing window first. To do this, press WINDOW and enter the following settings: Xmin = 60 (set X min =µ 4σ) Xmax = 140 (set X max =µ+4σ) Xscale= 5 (choose an appropriate value) Ymin = 0.01 (use a small negative value to show the x-axis ) Ymax = 0.05 (try 0.5 σ to start) Yscale= 0.01 (choose an appropriate value) Xres=1 (3) As a precaution, to avoid other graphs being plotted at the same time, go to (i) Y= and either CLEAR or deselect all functions. (ii) press 2nd [STATPLOT] and turn all plots OFF. (4) Return to the HOME screen ( 2nd [QUIT]) and press ENTER to complete the task. Thus, Pr(X 105) = area under curve = (correct to 4 decimal places). Note: Before proceeding to the next activity press 2nd [DRAW] ( PRGM ) ENTER to clear the drawing from the screen. Exercise 1. If X~N(60,5 2 ), display the following probabilities as an area under the normal curve: (1) Pr(54 X 58) (2) Pr(X 70) (3) Pr(X 62.5)

7 3.7 Continuous probability distributions K.3 Solving inverse normal problems Task: If X~N(168, 5 2 ), find the value of x such that: 1. Pr(X< x) = Pr(X x) = Pr(X x) = 0.05 Area = 0.95 µ=168!=5 Area = µ=168!=5 Area = 0.05 µ=168!=5 168 x=? 168 x=? x=? 168 The invnorm( command is located in the DISTR(ibution)menu under DISTR ( 2nd [DISTR] ( VARS )). Step 1. Find x such that Pr(X x) = 0.95, when X~N(168,5 2 ) Start on the HOME screen (press 2nd [QUIT] if you are not on the HOME screen), CLEAR. (1) Go to the DISTR menu ( 2nd [DISTR] ( VARS ) ) and use the arrow key to move down the DISTR menu to option 3: invnorm( and select by pressing ENTER. This pastes the invnorm( command onto the HOME screen. (2) Complete the command as follows: invnorm(0.95,168,5) probability µ! (3) Finally, press ENTER to evaluate. Thus, x = (correct to 1 decimal place).

8 3.8 Continuous probability distributions Step 2. Find x such that Pr(X x) = 0.025, when X~N(168,5 2 ). To proceed, we need to recognise that Pr(X x) = implies Pr(X<x) = and, as shown opposite, find x = (correct to 1 decimal place). Step 3. Find x such that Pr(X x) = 0.05, when X~N(168,5 2 ). Proceed directly as shown opposite to find x= (correct to 1 decimal place). Exercises 1. If Z~N(0, 1 2 ), find the value of z, correct to 2 decimal places, such that (1) Pr(Z z) = [z=1.00] (2) Pr(Z z) = [z= 2.00] (3) Pr(Z z)=0.95 [z=1.64] (4) Pr( Z z)=0.95 [z=1.96] 2. If X~N(10, 2 2 ), find the value of x, correct to 1 decimal place, such that (1) Pr(X x )= 0.05 [x=6.7] (2) Pr(X x) = 0.90 [x=7.4] (3) Pr(X x) =0.025 [x=13.9]

9 3.9 Continuous probability distributions K.4 Using the normal distribution to approximate a binomial distribution When n is large, the binomial distribution can be approximated by a normal distribution with mean µ=np and variance σ 2 = np(1-p). This can be seen by plotting the two distributions on the same graph. Plot 1 Comparing Bi(5,0.3) with N(1.5,1.05) (µ= np = 5 x 0.3=1.5, σ 2 = np(1-p) = 5 x 0.3 (1 0.3) = 1.05). Step 1. Construct a scatterplot displaying Bi(5, 0.3). See section H.2 for details. Step 2. On the same graph, plot the probability density function for N(1.5,1.05) as follows. (1) Open the Y= menu ( Y= ), deselect or clear all other entries, and opposite Y 1 =, paste in the normalpdf( command ( 2nd [DISTR] ( ENTER )). (2) Complete, as follows: normalpdf(x,1.5, (1.05)). (3) Press GRAPH to display. From the plot we see that the exact binomial probability for x=4 is , whereas the normal approximation normalcdf(3.5,4.5,1.5, (1.05))= an error of around 10%.

10 3.10 Continuous probability distributions Plot 2 Comparing Bi(25,0.3) with N(7.5,5.25) (µ= np = 25 x 0.3 = 7.5, σ 2 = np(1-p) = 25 x 0.3 (1 0.3) = 5.25). Step 1. Construct a scatterplot displaying Bi(25, 0.3). See section H.2 for details. Step 2. On the same graph plot the probability density function for N(7.5, 5.25) as follows. (1) Open the Y= menu ( Y= ), deselect or clear all other entries, and opposite Y 1 =, paste in the normalpdf( command ( 2nd [DISTR] ENTER ). (2) Complete, as follows: normalpdf(x,7.5, (5.25)). (3) Press GRAPH to display. From the plot we see that the normal curve more closely fits the binomial distribution as n increases. Comparing probabilities, we see that when X=4, for example, the exact binomial probability is , while the normal approximation normalcdf(3.5,4.5,7.5, (5.25))= an error of around 4%. This approximation will continue to improve as n increases. Exercise Graphically compare the normal approximation to the binomial for p=0.5 when n= 5 and n= 20. p=0.5, n= 5 p=0.5, n= 5 p=0.5, n= 20 p=0.5, n= 20

11 3.11 Continuous probability distributions L The exponential probability distribution A continuous random variable X is said to have an exponential distribution if it has a probability density function p(x) = λe -λx 0<x <. The exponential distribution is defined by a single parameter, λ. To indicate that a random variable has an exponential distribution with parameter λ, we write X ~ Exp(λ). For continuous probability distributions, probabilities are defined by areas under the probability density curve. Except in the case of the normal distribution, these areas are not programmed into the graphics calculator and must be determined from the graph. L.1 Determining and displaying exponential probabilities Task: Step 1. If X~ Exp(0.5), find Pr(1 X 4). Graph the distribution (1) Open the Y= menu ( Y= ), deselect or clear all other entries, and opposite Y 1 =, type in 0.5e -0.5x. (2) Go to the WINDOW menu WINDOW and enter the following settings: Xmin = 0 Xmax = 5 Xscale= 1 Ymin = 0.05 Ymax = 0.6 Yscale= 0.1 (3) Press GRAPH to plot. Step 2. Find the area under the curve To evaluate Pr(1 X 4), we need to find the area under the graph between x=1 and x=4. This is done from the graph through the CALC menu. Press 2nd [CALC] ( TRACE ) and select option 7: f(x)dx. Press ENTER and the calculator will ask you to specify the Lower Limit? type in 1 and press Enter. Upper limit? type in 4 and press Enter. The calculator will then calculate the required area, (correct to 4 decimal places) and shade it in on the graph. See opposite. Thus, if X ~ Exp(0.5), then Pr(1 X 4)=

12 3.12 Continuous probability distributions Exercise 1. If X ~ Exp(0.2), show that, correct to 4 decimal places: (1) Pr(3 X 5)= (2) Pr(X 5)= (3) Pr(X 2)=1-Pr(X 2)=