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EDA

1. Continuous Random Variable and Probability Distribution Prepared by: John Joko Banawa Marvin Tapao Jose Miguel Maulion Ronan Bruna John Justine Fermalan Ryan Jester Famorcan Juliana Marie Galon Najievah Forio John Carl Mameng

2. Recap Determine if the given example is discrete or continuous random variable. Number of defective light bulbs in a box of ten . Tossing a coin . Time . Height. Weight. Number of books you read in the library. Amount of sugar in an orange. Rolling a dice. Population in Roxas, Oriental Mindoro. Temperature.

3. What is a Continuous Random Variable? A continuous random variable can be defined as a random variable that can take on an infinite and uncountable set of positive values known as range. Due to this, the probability that a continuous random variable will take on an exact value is 0. The probability density function and the cumulative distribution function are used to describe the characteristics of a continuous random variable.

4. Probability density function of continuous random variable Recall that continuous random variables have uncountably many possible values. Just as for discrete random variables, we can talk about probabilities for continuous random variables using density functions. The probability density function, denoted f, of a continuous random variable X satisfies the following: f is piecewise continuous  

5. The first three conditions in the definition state the properties necessary for a function to be valid probability density function for a continuous random variable. The fourth condition tells us how to us a probability density function to calculate probabilities for continuous random variables, which are given by integrals the continuous analog to sums. Let X be a continuous random variable. Then a probability distribution or probability density function of X is a function f(x) such that for any two numbers a and b with , we have:  

6. The probability that X is in the interval [ a, b] can be calculated by integrating the probability density function of the random variable X. The probability that X takes on a value in the interval [a, b] is the area above this interval and under the graph of the density function:

7. Clearly How can we show that the area of this probability density distribution is equal to 1? How do we calculate   Consider the reference line connecting the valve stem on a tire to the center point. Let X be the angle measured clockwise to the location of an imperfection. One possible probability density function for X is      

8. Example: Let the random variable X denote the time a person waits for an elevator to arrive. Suppose the lonest one would need to wait for the elevator is 2 minutes, so that the possible values of X(in minutes) are given by the interval [0, 2]. A possible probability density distribution for X is given by: for , for otherwise  

9. So, if we wish to calculate the probability that a person waits less than 30 seconds(or 0.5 minutes) for the elevator to arrive, then we calculate the following probability using the probability density function:

10. Cumulative distribution function The  cumulative distribution function  of a continuous random variable can be determined by integrating the probability density function. It can be defined as the probability that the random variable, X, will take on a value that is lesser than or equal to a particular value, x. The formula for the cumulative distribution function of a continuous random variable, evaluated between two points a and b, is  

11. Example: 1. The cumulative distribution function for the random variable X is given by: Calculate       2. The cumulative distribution function for the random variable X is given by: Calculate .      

12. Mean and Variance of Continuous Random Variable Mean of Continuous Random Variable- The  mean  of a continuous random variable can be defined as the  weighted average  value of the random variable, X. It is also known as the expectation of the continuous random variable. The formula is given as  

13. Example: The random variable X has a probability density function given by: Find the mean of the continuous random variable. The probability density function for the random variable X is given by: Find the mean.   otherwise    

14. Variance of Continuous Random Variable- The  variance  of a continuous random variable can be defined as the expectation of the squared differences from the mean. It helps to determine the dispersion in the distribution of the continuous random variable with respect to the mean. The formula is given as   

15. Standard Deviation(   Is a measure of how dispersed the data is in relation to the mean. Low, or small, standard deviation indicates data are clustered tightly around the mean, and high, or large, standard deviation indicates data or more spread out.

16. Example: The random variable X has a probability density function given by: Find the mean and variance of X. The probability density function for the random variable X is given by: Find the mean and standard deviation of X.   otherwise    

17. Probability Distributions of Continuous Random Variable A continuous random variable is usually used to model situations that involve measurements. For example, the possible values of the temperature on any given day. As the temperature could be any real number in a given interval thus, a continuous random variable is required to describe it. Some important continuous random variables associated with certain probability distributions are given below. Uniform Random Variable Normal Random Variable Exponential Random Variable

18. Continuous Uniform Random Variable A continuous random variable that is used to describe a uniform distribution  is known as a uniform random variable. Such a distribution describes events that are equally likely to occur, all outcomes are equally possible . Each variable has the same chance of being hit as a result. Random variables are spaced evenly in this symmetric probabilistic distribution, with a probability. The probability density function of a uniform random variable is as follows :   otherwise

19. Example: Bus is uniformly late between 2 and 10 minutes. How long can you expect to wait? With what Standard Deviation? If it’s >7 minutes late, you’ll be late for your class in Engineering Data A nalysis subject. What’s the probability of you being late? The amount of time that it takes a student to complete a chemistry test is uniformly distributed between 20 and 45 minutes. What is the probability that a student will take more than 36 minutes to complete the exam?

20. Normal Distribution Normal Distribution is one of the most basic continuous distribution types. It is probably the most important distribution in all of probability and statistics . Around its mean value, this probability distribution is symmetrical. It also demonstrates that data close to the mean occurs more frequently than data far from it. Here, the mean is 0, and the variance is a finite value.

21. Example: A 100-watt light bulb has an average brightness of 1640 lumens, with a standard deviation of 62 lumens. What is the probability that a 100-watt light bulb will have a brightness more than 1800 lumens? What is the probability that a 100-watt light bulb will have a brightness less than 1550 lumens?

22. Exponential Random Variable The exponential distribution describes distance(or length, area, volume) between events described by a Poisson process(discrete events occurring on a continuous interval). In the exponential distribution, the random variable, X, is the interval between two events. It is a continuous random variable and can be described by: Oftentimes, we are interested in the cumulative exponential distribution function: Mean Variance  

23. Example 1: In the last 40 years, there have been 200 earthquakes. What is the probability that there will be at least 3 months until the next earthquake?

24. Example 2: Laptops are produced by JJ tech. last, on average, for 5 years. The life span of each laptop follows an exponential distribution. Calculate the rate parameter. Write the probability density function and graph it. what is the probability that a laptop will last less than 3 years? What is the probability that a laptop will last mmore than 10 years? What is the probability that a laptop will last between 4 and 7 years?

25. Activity: Part I: Directions : Rearrange the jumbled letters and fill the missing puzzle with the corresponding words from the jumbled letters in each item . ACROSS 2. A _______ random variable can be defined as a random variable that can take on an infinite and uncountable set of positive values . UUNTCSOINO 3. ______ distribution is one of the most basic continuous distribution types. It is probably the most important distribution in all of probability and statistics. MLOARN 4. Such a distribution describes events that are equally likely to occur, all outcomes are equally possible. Each variable has the same chance of being hit as a result. IMORFUN 7. The ________ distribution describes between events described by a Poisson process. PTNEAOXENLI 8. __________ can be defined as the expectation of the squared differences from the mean. It helps to determine the dispersion in the distribution of the continuous random variable with respect to the mean. RACEAVNI DOWN 1. The  _________ distribution function of a continuous random variable can be determined by integrating the probability density function. It can be defined as the probability that the random variable, X, will take on a value that is lesser than or equal to a particular value x. MAEUCULVIT 5. ________ can be defined as the weighted average value of the random variable, X. It is also known as the expectation of the continuous random variable. NAME 6. The probability ________ function of a continuous random variable can be defined as a function that gives the probability that the value of the random variable will fall between a range of values. YSEDINT

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27. Part II: Problem Solving 1. If the probability density function is give as Find 2. Ronan toss a coin twice. Let X be the number of observed heads. Find the cumulative distribution function of X. 3. The probability density function of a continuous random variable, X, is given as follows. Find   x   X+3   otherwise

28. Answer Key: Part I: Cumulative Continuous Normal Uniform Mean Density Exponential Variance Part II: 5/6 ¾ 0.125

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