A LEVEL MATHS: Probability Distributions

What is the shortcut for Binomial Probability Function on GDC?

--> menu-->5: Probability -->5: Distributions --> A: Binomial Pdf

What is the shortcut for Binomial Cumulative Function?

--> menu-->5: Probability -->5: Distributions --> B: Binomial Cdf

When asked to estimate the parameters of normally distributed data and you are provided with a graph, what process should you take?

-Look for the peak of the bell curve and read the x axis to find out what µ= -To calculate σ, recall that at x=µ±σ is where the points of inflections are located on the bell curve -Find the points of inflection and from that estimate where µ±σ and then calculate σ from there

What is µ equal to on a standard normal distribution?

0

Label the mean on a normall distribution graph

µ = MEAN

On a normal distribution graph what does P(X≥µ) =

0.5 This is the probability that X is less than or equal to µ. Given that µ (the mean) is in the middle of the graph then you half the total area beneath the graph which is 1/2= 0.5 (or find the limit at where the mean is and where the graph begins)

How many ways can 4 different ornaments be arranged?

4!= 24

How many ways can 5 different objects be arranged if 2 of those objects are identical?

5!/2! =60

How many ways can 7 different objects be arranged if 4 of those objects are identical?

7!/4! = 210

How many ways can 8 different ornaments be arranged?

8!= 40 320

Sketch a normal distribution curve with a small variance (σ²)

Blue curve reresents small variance

What is the formula which you use to convert Normal Distribution to Standard Normal Distirbution ?

If X~N(µ,σ²), then, (X-µ)/σ = Z, where Z~N(0,1)

What does the Inverse normal function tell you?

It uses the probability to tell you the range of values for X where the probability of X falling in the range is a give n probability.

What is a probaility function?

This is a formula which generates the probalities for different values of x

What is 'x'?

This is a particular value that X (uppercase) can take.

What is a probability distribution?

This is a table showing the possible values of lower case x from the X variable

What is 'X'?

This is the name of a random variable

I have an unfair coin. When I toss this coin the probability of getting heads is 0.35. Find the probability that it will land on heads fewer than 3 times when I toss it 12 times in total

Use the Binomial Cdf:

I roll a fair six-sided dice 5 times. Find the probability of rolling: a) 2 sixes b) 3 sixes c) 4 numbers less than 3

Use the binomial probability function a) 0.161 b) 0.032 c) 0.041

find the value of µ given that X~N(µ,2²) and P(X<23) = 0.9015

When you are asked to find µ 1) First you need to do a standardisation using the formula Z= (X-µ)/σ 2) Then you type the probability/ area into inverse normal distribution 3) Then you type in the µ=0 and σ=1 because we have converted to standard normal distribution 4) Then you can set up an equation to find µ

Find the value of σ given that X~N(53, σ²) and P(X<50) =0.1. Find σ

When you are asked to find σ 1) First you need to do a standardisation using the formula Z= (X-µ)/σ 2) Then you type the probability/ area into inverse normal distribution 3) Then you type in the µ=0 and σ=1 because we have converted to standard normal distribution 4) Then you can set up an equation to find σ

What is the formula which you can use for approximating binomial distributions with normal distributions?

X ~B(n,p) approximation Y ~ N(np, npq) (where q= 1-p) Also remember that when you are carrying out an approximation with a ragne e.g. (6<X<7), because you are approximating a discrete quatnity using a continuous variable you must change it to (5.5<X<7.5)

How do you calculate what the significance level of a two-tailed test is?

You add together both probabilities together

When you are finding the area beneath the curve on a normally distributed graph and you are not provided with a lower bound / upper bound e.g. P(X≤17) or P(X≥4)?

You create a lower bound/ upper bound which is really small/ really big e.g. -9999/ 9999

When you are given questions about the inverse normal function/ normal cdf/ binomial cdf and there is a ≥ or a >, e.g. P(X>a)= 0.2418 what must you do?

You must do 1-p 1-0.2418 = P(X<a) and you need it in this form if yo are to find the value of a

Write out the distribution representation for standard normal distribution

Z~N(0,1) This means that 0= the mean and 1=the variance

What are the two ways of representing a Distribution: a) Binomial: X~..... b) Normal: X~.....

a) X~ (n,p) (n= number of trials, p= probability of success) b) X~(µ,σ²) (µ= mean, σ²= variance)

The weights of bags of popcorn are normally distributed with mean of 200 g and 60% of all bags weighing between 190 g and 210 g. (a) Write down the median weight of the bags of popcorn. (b) Find the standard deviation of the weights of the bags of popcorn.

a)We know that the mean =200, therefore the median in this case is also 200, as it is directly in the middle b) Draw the graph for normal distribution with 200 in the middle of the graph. Acknowledge that p(190< X < 210)= 0.6 and that P(200<X<210) = 0.3 Therefore P(X<210) = 0.5 + 0.3 = 0.8 P(X<210) - P(X<190) = 0.6 0.8 - 0.6= P(X<190) P(X<190) =0.2 Then do the inverse normal cdf and also do the standardisation -10/σ = -0.841... σ=11.88

On a normal distribution graph, what does P(X≤µ)=

0.5 This is the probability that X is MORE than or equal to µ. Given that µ (the mean) is in the middle of the graph then you half the total area beneath the graph which is 1/2= 0.5 (or find the limit at where the mean is and where the graph begins)

What is the area under a normal distribution graph equal to? What does this indicate?

1 That the total probability is 1

For a discreet random variable, what does ∑P(X=x)=?

1 It states that when you add up all of the probabilities, you end up with 1

What are the 4 conditions for a binomial distribution?

1) Data is discreet 2) data represents the number of successes in a fixed number (n) of trials 3) All the trials are independent 4) The probability of 'success' (p) is the same in each trial If this is all true then: X~B(n,p) P(X=x)= (n over x) x pˣ x (1-p)ⁿ⁻ˣ Use this to comment on the appropriateness of the binomial model

Find the value of both µ and σ given that X~N(µ,σ²). If P(X<9)= 0.5596 and P(X>14)= 0.0322

1) First you to to do a standardisation using the formula Z = (X-µ)/σ, twice 2) Then you type the the probability/ area into inverse normal distribution. You have to be careful with P(X>14)= 0.0322. You have to put it in the form P(X<14)=1-0.0322

When you are provided with the probability of a normal distribution e.g. x~n(28, 36). If P(a< X< 30) = 0.5842 how do you calculate a?

1) SKETCH OUT GRAPH! Write out that µ=28 then write out that σ=6 2) Then you write out that P(X<30) - P(a<X) = 0.5842 3) Then use the GDC to calculate P(X<30) using normal CDF= 0.630559 4) Set up equation so it shows 0.630559-0.5842= P(a<X) P(a<X)=0.046359 5) Then You open up you GDC and you go to -> menu -> 5: probability -> 5: Distributions -> 3: Inverse Normal function 3) Then you type in the probability which is equal to the area. Then you type in the mean and the standard deviation 4) This will give you an answer of 17.9126

When given that P(X≤/≥/>/<...) for a variable which is normally distributed, how could you use you GDC to calculate the area under the bell curve which you are trying to find? e.g. X~(12, 16) Find: P(X≤17)

1) Sketch the bell curve and the area beneath it which you want to find 2) You have beem provided with the variance (16)- to make that the standard deviation, square rt it to get 4. 3) Aquire your GDC and type in ->menu -> 5) Probabbility -> 5) Distributions -> 2) Normal Cdf 4) For you lower bound make it an incredibly large number e.g. -9999 and for your upper bound inser 17. For µ insert your mean and for σ insert 4. 5) Your answer should be 0.894

What are the 3 conditions for Normal Distribtutions?

1) The data is continuous 2) The data is symmetrically distributed with a peak in the middle (µ=mean) 3)The data tails off on either side of the mean (virtually all of the data is within 3 standard deviations of the mean) If these conditions are met then X~N(µ,σ²) Use this when finding a suitable distribution

When given the question: If X~N(5, 16), find P(5 < X < 11) by first converting to standard normal distribution

1) The you have standardised your range to form new limits, you then open up you calculator and to normal CDF 2)Type in the new limits and insert µ=0 and σ=1

When you are provided with the probability of a normal distribution e.g. x~n(85, 25). If P(X<a) = 0.9192 how do you calculate a?

1) Write out that µ=85 then write out that σ=5 2) You open up you GDC and you go to -> menu -> 5: probability -> 5: Distributions -> 3: Inverse Normal function 3) Then you type in the probability which is equal to the area. Then you type in the the mean and the standard deviation 4) This will give you an answer of 91.99

Sketch a normal distribution curve with a large variance (σ²)

Red line represents large variance

What proportion of the total area under a normal distribution graph lies within three standard deviations (µ±3σ) of the mean?

almost all

What proportion of the total area under a normal distribution graph lies within one standard deviation (µ±σ) of the mean?

approximately 2/3

What proportion of the total area under a normal distribution graph lies within two standard deviations (µ±2σ) of the mean?

approximately 95%

Where are the x-coordinates for the points of inflection on a normally distributed curve? Sketch a normal distribution curve and label these points of inflection

at x=µ+σ and x=µ-σ

Using the information below- complete the probaility distribution by finding the value of a: a) The discrete random variable X, where X can only take values 0, 1, 2, 3, 4, has the probability shown below 0.1, 0.2,0.3, 0.2, a b) Find P(2≤X<4)

b) 0.5

What are the conditions under which you can use a normal approximation for a Binomial distribution?

if p=0.5 (or is close to 0.5 like 0.5) if n= large (e.g. n=800) if np and nq are larger than 5 then the normal approximation usually works fine, even if p is not very close to 0.5

Using the information below- write out the the probability distribution: a)The random variable X, where X can only take values of 1,2,3 has probaility function P(X=x) = kx b)Find the value of K

k= 1/6