Note: please follow the steps in the photo attached B. Variance and Standard Deviation of a Discrete Random VariableThe variance of a random variable is the expected value of the square of the difference betweenthe assumed value of random variable and the mean. It is denoted by ² or Var (X).Var (X)=[(x-µ)² P(x)] or o² = (x-μ)² P(x)]Where x = outcome, μ = population mean, P(x) =probability of the outcomeThe standard deviation of a random variable denoted by o is the square root of the variance.0 =Σ[(x − µ)² P(x)]Note: The larger the value of the variance or standard deviation, the farther are the values of Xfrom the mean. It means that the larger the value of the variance or standard deviation, the morespread are the values X from the mean or expected value.Example:A researcher surveyed the households in a small town. The random variable X representsthe number of college graduates in the households. The probability distribution of X is shownbelow. Determine the variance and standard deviation.X02P(X=x)0.250.500.25Solution:Step 1. Find the expected value.Based from previous example, E(X) or μ =1.Step 2. Subtract the expected value from each outcome. Square each difference.Step 3. Multiply each squared difference by the corresponding probability.Step 4. Sum up all the figures obtained in step 3.XP(x)x P(x)x-μ(x-μ)² P(x)(x-μ)²100.2500-1 -11(0.25) = 0.2510.500.501-1=000(0.50)=020.250.502-1=111(0.25) = 0.25[x P(x)] = 1.000² =[(x-μ)² P(x)] = 0.50Variance, o² = 0.50(You just need to take the square root of the value of variance to compute the standard deviation)Standard Deviation, o = √0.50 ≈ 0.71Based from the standard deviation it is evident that the variance or standard deshows a low measure of spread which is associated with the random variable Xcorresponding probabilities. In other words, the possible values x and their corresprobabilities are close to each other.Take note that it is easier to interpret the value of standard deviation because it usessame unit of measure of the random variable X. III. Practice ExercisesA random variable X given in the probability distribution below is the number of persons enteringthe bank every hour during a span of one week which were recorded by the security guard. a) What isthe expected number of persons entering the bank every hour during that week? b) Compute the varianceand standard deviation by illustrating its calculation step by step.1x234P(x)0.100.200.450.25IV Evaluation

Question

Accepted Answer

The random variable X denotes the number of persons entering the bank every hour during a span of…