Mean or Expected Value and Standard Deviation Quiz

PDF page 249: The expected value of a discrete random variable X, symbolized as E(X), is often referred to as the long-term average or mean (symbolized as μ). This means that over the long term of doing an experiment over and over, you would expect this average. For example, let X = the number of heads you get when you toss three fair coins. If you repeat this experiment (toss three fair coins) a large number of times, the expected value of X is the number of heads you expect to get for each three tosses on average.

PDF page 249: To find the expected value, E(X), or mean μ of a discrete random variable X, simply multiply each value of the random variable by its probability and add the products. The formula is given as

PDF page 249: Here x represents values of the random variable X, P(x) represents the corresponding probability, and symbol represents the sum of all products xP(x). Here we use symbol μ for the mean because it is a parameter. It represents the mean of a population.

PDF page 249: Here x represents values of the random variable X, P(x) represents the corresponding probability, and symbol represents the sum of all products xP(x). Here we use symbol μ for the mean because it is a parameter. It represents the mean of a population.

PDF page 249: The expected value/mean is 1.1. The men's soccer team would, on the average, expect to play soccer 1.1 days per week.

PDF page 250: The number 1.1 is the long-term average or expected value if the men's soccer team plays soccer week after week after week.

PDF page 249: Value Table This table is called an expected value table. The table helps you calculate the expected value or longterm average.

PDF page 250: As you learned in Chapter 3, if you toss a fair coin, the probability that the result is heads is 0.5. This probability is a theoretical probability, which is what we expect to happen. This probability does not describe the short-term results of an experiment. If you flip a coin two times, the probability does not tell you that these flips will result in one head and one tail. Even if you flip a coin 10 times or 100 times, the probability does not tell you that you will get half tails and half heads. The probability gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24,000 times! He recorded the results of each toss, obtaining heads 12,012 times. The relative frequency of heads is 12,012/24,000 = .5005, which is very close to the theoretical probability .5. In his experiment, Pearson illustrated the law of large numbers. The law of large numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero (the theoretical probability and the relative frequency get closer and closer together). The relative frequency is also called the experimental probability, a term that means what actually happens. In the next example, we will demonstrate how to find the expected value and standard deviation of a discrete probability distribution by using relative frequency. Like data, probability distributions have variances and standard deviations. The variance of a probability distribution is symbolized as and the standard deviation of a probability distribution is symbolized as σ. Both are parameters since they summarize information about a population. To find the variance of a discrete probability distribution, find each deviation from its expected value, square it, multiply it by its probability, and add the products. To find the standard deviation σ of a probability distribution, simply take the square root of variance . The formulas are given as below.

PDF page 250: As you learned in Chapter 3, if you toss a fair coin, the probability that the result is heads is 0.5. This probability is a theoretical probability, which is what we expect to happen. This probability does not describe the short-term results of an experiment. If you flip a coin two times, the probability does not tell you that these flips will result in one head and one tail. Even if you flip a coin 10 times or 100 times, the probability does not tell you that you will get half tails and half heads. The probability gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24,000 times! He recorded the results of each toss, obtaining heads 12,012 times. The relative frequency of heads is 12,012/24,000 = .5005, which is very close to the theoretical probability .5. In his experiment, Pearson illustrated the law of large numbers. The law of large numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero (the theoretical probability and the relative frequency get closer and closer together). The relative frequency is also called the experimental probability, a term that means what actually happens. In the next example, we will demonstrate how to find the expected value and standard deviation of a discrete probability distribution by using relative frequency. Like data, probability distributions have variances and standard deviations. The variance of a probability distribution is symbolized as and the standard deviation of a probability distribution is symbolized as σ. Both are parameters since they summarize information about a population. To find the variance of a discrete probability distribution, find each deviation from its expected value, square it, multiply it by its probability, and add the products. To find the standard deviation σ of a probability distribution, simply take the square root of variance . The formulas are given as below.

PDF page 250: As you learned in Chapter 3, if you toss a fair coin, the probability that the result is heads is 0.5. This probability is a theoretical probability, which is what we expect to happen. This probability does not describe the short-term results of an experiment. If you flip a coin two times, the probability does not tell you that these flips will result in one head and one tail. Even if you flip a coin 10 times or 100 times, the probability does not tell you that you will get half tails and half heads. The probability gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24,000 times! He recorded the results of each toss, obtaining heads 12,012 times. The relative frequency of heads is 12,012/24,000 = .5005, which is very close to the theoretical probability .5. In his experiment, Pearson illustrated the law of large numbers. The law of large numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero (the theoretical probability and the relative frequency get closer and closer together). The relative frequency is also called the experimental probability, a term that means what actually happens. In the next example, we will demonstrate how to find the expected value and standard deviation of a discrete probability distribution by using relative frequency. Like data, probability distributions have variances and standard deviations. The variance of a probability distribution is symbolized as and the standard deviation of a probability distribution is symbolized as σ. Both are parameters since they summarize information about a population. To find the variance of a discrete probability distribution, find each deviation from its expected value, square it, multiply it by its probability, and add the products. To find the standard deviation σ of a probability distribution, simply take the square root of variance . The formulas are given as below.

PDF page 250: As you learned in Chapter 3, if you toss a fair coin, the probability that the result is heads is 0.5. This probability is a theoretical probability, which is what we expect to happen. This probability does not describe the short-term results of an experiment. If you flip a coin two times, the probability does not tell you that these flips will result in one head and one tail. Even if you flip a coin 10 times or 100 times, the probability does not tell you that you will get half tails and half heads. The probability gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24,000 times! He recorded the results of each toss, obtaining heads 12,012 times. The relative frequency of heads is 12,012/24,000 = .5005, which is very close to the theoretical probability .5. In his experiment, Pearson illustrated the law of large numbers. The law of large numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero (the theoretical probability and the relative frequency get closer and closer together). The relative frequency is also called the experimental probability, a term that means what actually happens. In the next example, we will demonstrate how to find the expected value and standard deviation of a discrete probability distribution by using relative frequency. Like data, probability distributions have variances and standard deviations. The variance of a probability distribution is symbolized as and the standard deviation of a probability distribution is symbolized as σ. Both are parameters since they summarize information about a population. To find the variance of a discrete probability distribution, find each deviation from its expected value, square it, multiply it by its probability, and add the products. To find the standard deviation σ of a probability distribution, simply take the square root of variance . The formulas are given as below.

PDF page 250: Here x represents values of the random variable X, μ is the mean of X, P(x) represents the corresponding probability, and symbol ∑ represents the sum of all products To find the standard deviation, σ, of a discrete random variable X, simply take the square root of the variance .

PDF page 250: As you learned in Chapter 3, if you toss a fair coin, the probability that the result is heads is 0.5. This probability is a theoretical probability, which is what we expect to happen. This probability does not describe the short-term results of an experiment. If you flip a coin two times, the probability does not tell you that these flips will result in one head and one tail. Even if you flip a coin 10 times or 100 times, the probability does not tell you that you will get half tails and half heads. The probability gives information about what can be expected in the long term. To demonstrate this, Karl Pearson once tossed a fair coin 24,000 times! He recorded the results of each toss, obtaining heads 12,012 times. The relative frequency of heads is 12,012/24,000 = .5005, which is very close to the theoretical probability .5. In his experiment, Pearson illustrated the law of large numbers. The law of large numbers states that, as the number of trials in a probability experiment increases, the difference between the theoretical probability of an event and the relative frequency approaches zero (the theoretical probability and the relative frequency get closer and closer together). The relative frequency is also called the experimental probability, a term that means what actually happens. In the next example, we will demonstrate how to find the expected value and standard deviation of a discrete probability distribution by using relative frequency. Like data, probability distributions have variances and standard deviations. The variance of a probability distribution is symbolized as and the standard deviation of a probability distribution is symbolized as σ. Both are parameters since they summarize information about a population. To find the variance of a discrete probability distribution, find each deviation from its expected value, square it, multiply it by its probability, and add the products. To find the standard deviation σ of a probability distribution, simply take the square root of variance . The formulas are given as below.