Continuous Random Variables & Probability Distributions

Continuous Random Variables & Probability Distributions

What is a Random Variable? It is a quantity whose values are real numbers and are determined by the number of desired outcomes of an experiment. Is there any special Random Variable? We can categorize random variables in two groups: Discrete random variable Continuous random variable

What are Discrete Random Variables? It is a numerical value associated with the desired outcomes and has either a finite number of values or infinitely many values but countable such as whole numbers 0,,2,3,. What are Continuous Random Variables? It has infinitely numerical values associated with any interval on the number line system without any gaps or breaks.

What is a Probability Distribution? It is a description and often given in the form of a graph, formula, or table that provides the probability for all possible random variables of the desired outcomes. Are there any Requirements? Let x be any random variable and P(x) be the probability of the random variable x, then P(x) = 0 P(x)

What is Discrete Probability Distributions? It is a probability distribution for a discrete random variable x with probability P(x) such that P(x) =, and 0 P(x). Here are some examples of continuous probability distribution: Discrete Probability Distribution Binomial Probability Distribution

What is Continuous Probability Distributions? It is a probability distribution for a continuous random variable x with probability P(x) such that P(x) =, 0 P(x), and P(x = a) = 0. Here are some examples of continuous probability distribution: Uniform Probability Distribution Standard Normal Probability Distribution Normal Probability Distribution

What is Uniform Probability Distributions? It is a probability distribution for a continuous random variable x that can assumes all values on the interval [a,b] such that All values are evenly spread over the interval [a,b], The graph of the distribution has a rectangular shape, and P(c < x < d) = (d c) as displayed below. b a b a a c d b

Example: The amount of time, in minutes, that a person must wait for a bus is uniformly distributed between 0 and 2 minutes, inclusive. Draw and label the uniform probability distribution, Find the probability that a randomly selected person has to wait for a bus is between 7.5 to 0 minutes? Find k = P 90, and explain what this number represents. Solution: We have a uniform probability distribution with a = 0, b = 2, and a rectangular graph with the width of b a = 2

Continued Solution: Draw and label the uniform probability distribution, 2 0 2 Find the probability that a randomly selected person has to wait for a bus is between 7.5 to 0 minutes? We need to redraw our uniform probability distribution, label, and shade a region that represents the probability for wait time from 7.5 to 0 minutes.

Continued Solution: 2 0 7.5 0 2 Now to the find the probability, P(7.5 < x < 0) = (0 7.5) = 2.5 2 = 5 24 0.208 2

Continued Solution: Find k = P 90, and explain what this number represents. The gray area below is 0.9 since it represents k = P 90, 2 0.9 0 k 2 P(x < k) = (k 0) 2 0.9 = k 2 k = 0.8 So 90% of the customers have a wait time that is below 0.8 minutes.

Finding Mean, Variance, and Standard Deviation of a Uniform Probability Distribution: Given a continuous random variable x that can assumes all values on the interval [a, b] with a uniform probability distribution, then Mean µ = b +a 2, Variance σ 2 (b a)2 =, and 2 Standard Deviation σ = σ 2.

Example: A continuous random variable x that assumes all values from x = 5 and x = 30 with uniform probability distribution. Draw and label the uniform probability distribution, Find the mean of the distribution. Find the variance of the distribution. Find the standard deviation of the distribution. Solution: We have a uniform probability distribution with a = 5, b = 30, and a rectangular graph with the width of b a = 25.

Continued Solution: A continuous random variable x that assumes all values from x = 5 and x = 30 has a uniform probability distribution. Draw and label the uniform probability distribution, 25 5 30 Find the mean of the distribution. µ = b +a 2 = 30+5 µ = 7.5 2

Continued Solution: Find the variance of the distribution. σ 2 = (b a)2 2 = (30 5)2 2 σ 2 = 625 2 Find the standard deviation of the distribution. σ = σ 2 625 = 2 σ 7.27

Example: The amount of coffee dispensed by a certain machine into a cup is a continuous random variable x that assumes all values from x = and x = 25 ounces with uniform probability distribution. Draw and label the uniform probability distribution, Find the probability that a cup filled by this machine will contain at least 4.5 ounces. Find the probability that a cup filled by this machine will contain at most 20 ounces. Find the probability that a cup filled by this machine will contain more than 6.5 ounces but less than 8 ounces. Find k such that P(x > k) = 0.3. Explain what this number represents.

Solution: In this example, we have a uniform probability distribution with a =, b = 25, and a rectangular graph with the width of b a = 4 Draw and label the uniform probability distribution, 4 25 Find the probability that a cup filled by this machine will contain at least 4.5 ounces. P(x 4.5) = P(x > 4.5) = (25 4.5) 4 = 0.75

Continued Solution: Find the probability that a cup filled by this machine will contain at most 20 ounces. P(x 20) = P(x < 20) = (20 ) 4 0.643 Find the probability that a cup filled by this machine will contain more than 6.5 ounces but less than 8 ounces. P(6.5 < x < 8) = (8 6.5) 4 0.07 Find k such that P(x > k) = 0.3. Explain what this number represents. (25 k) = 0.3 k = 20.8 4 The probability that the machine will fill a cup more than 20.8 ounces in 0.3.

What is Standard Normal Distributions? It is a probability distribution for a continuous random variable z that can assumes any values such that The graph of the distribution is symmetric, and bell-shaped, The total are under the curve is equal to, Mean, mode, and median are all equal, µ = 0 and σ =, and P(a < z < b) is the area under the curve on the interval (a,b). P(a < z < b) a µ = 0 σ = b

Example: Find P(.5 < z < ). Solution: We start by drawing the normal curve, then shade and label accordingly. P(.5 < z < ).5 µ = 0 σ = Now we can use the normal distribution chart or different technology to compute the area which is the probability that we are looking for. In this case, P(.5 < z < ) = 0.7745.

Example: Find P(z >.5). Solution: We start by drawing the normal curve, then shade and label accordingly. P(z >.5) µ = 0 σ =.5 Now we can use the normal distribution chart or different technology to compute the area which is the probability that we are looking for. In this case, P(z >.5) = 0.0668.

Example: Use any method that you prefer to find the following and very your results with the drawing given below. P(0 < z < ) 2 P( 2 < z < 2) 3 P(z > ) 4 P(z < 2)

How do we find Z Score from Known Area? It is a probability distribution for a continuous random variable z that can assumes any values such that Draw a bell-shaped curve, Clearly identify and shade the region that represents the known area, Working with the cumulative area from the left, use the normal distribution chart or different technology to find the corresponding Z score. Needed Area Z µ = 0 σ =

Example: Use any method that you prefer to find k such that P(z > k) = 0.85. Solution: Needed Area 0.5 Known Area 0.85 k µ = 0 σ = Using any method, we get k =.036, that is P(z >.036) = 0.85 and P(z <.036) = 0.5. We can also conclude that k = P 5 =.036.

What is Z α? It is a notation that describes a z score with an area α to its right. How do we find Z α? We use the fact that Z α = Z α. Draw a bell-shaped curve, clearly identify, and shade the regions that represent the area for α, and α, Working with the cumulative area from the left, use the normal distribution chart or different technology to find the corresponding Z score. α α µ = 0 σ = Z α

Example: Use any method that you prefer to find Z 0.025. Solution: Given Z 0.025, we have α = 0.025, and α = 0.975, so we draw and label our normal curve. 0.975 0.025 µ = 0 σ = Z 0.025 Using any method, we get Z 0.025 =.960.

What is Normal Distributions? It is a probability distribution for a continuous random variable x that can assumes any values such that The graph of the distribution is symmetric, and bell-shaped, The total are under the curve is equal to, Mean, mode, and median are all equal, µ and σ are given, and P(a < x < b) is the area under the curve on the interval (a,b). P(a < x < b) a µ σ b

Example: Consider a normal distribution with the mean of 5 and standard deviation of 0. Find P(00 < x < 30). Solution: We start by drawing the normal curve, then shade and label accordingly. P(00 < x < 30) 00 µ = 5 30 σ = 0 Now we can use different technology to compute the area which is the probability that we are looking for. In this case, P(00 < x < 30) = 0.8664.

Example: Consider a normal distribution with the mean of 75 and standard deviation of 7.5. Find P(x > 90). Solution: We start by drawing the normal curve, then shade and label accordingly. P(x > 90) µ = 75 σ = 7.5 Now we can use different technology to compute the area which is the probability that we are looking for. In this case, P(x > 90) = 0.0228. 90

Example: Consider a normal distribution with the mean of 6375 and standard deviation of 200. Find P(x < 6000). Solution: We start by drawing the normal curve, then shade and label accordingly. P(x < 6000) 6000 µ = 6375 σ = 200 Now we can use different technology to compute the area which is the probability that we are looking for. In this case, P(x < 6000) = 0.0304.

Example: Consider a normal distribution with the mean of 25 and standard deviation of 2. Find x = P 80. Solution: 0.8 0.2 µ = 25 σ = 2 P 80 Using any method, we get x = P 80 = 35., that is P(x < 35.) = 0.85 and P(x > 35.) = 0.2.

Normal Probability Distributions & TI Normal Distribution P(a < x < b) P(x < a) P(x > a) P k Z α TI Command normalcdf(a,b,µ,σ) normalcdf( E99,a,µ,σ) normalcdf(a,e99,µ,σ) invnorm(k%,µ,σ) invnorm( α,0,)