MEAN, VARIANCE AND STANDARD DEVIATION OF A DISCRETE RANDOM VARIABLE

MEAN, VARIANCE AND STANDARD DEVIATION OF A DISCRETE RANDOM VARIABLE.

MEAN OF A DISCRETE RANDOM VARIABLE. MEAN- is the value that we would expect to observe on average if the experiment is repeated many times..

MEAN OF A DISCRETE RANDOM VARIABLE. The mean (also called the expected value ) of a discrete random variable X is the number µ = ?[X·P(X)] where: µ = mean; X = values of the random variable X; and P(X) = the corresponding probabilities..

. Example. Consider rolling a die. What is the average number of spots that would appear?.

. SOLUTION. Step 1 . Construct the probability distribution for the random variable X representing the number of spots that would appear..

. SOLUTION. Step 2 . Multiply the value of the random variable X by the corresponding probability..

. SOLUTION. Step 3 . Add the results obtained in Step 2..

. SOLUTION. Step 4 . Substitute the values into the formula. µ = ?[X·P(X)] µ = 7/2 or 3.5 Therefore, the mean or the expected value of the probability distribution is 7/2 or 3.5.

. Example. The probabilities that a surgeon operates on 2, 3, 4, 5 or 6 patients in any day are 0.20, 0.10, 0.20, 0.20 and 0.30, respectively. Find the mean of patients that a surgeon operates on a day..

. SOLUTION. Step 1 . Construct the probability distribution for the random variable Y representing the number of patients that a surgeon operates on a day..

. SOLUTION. Step 2 . Multiply the value of the random variable Y by the corresponding probability..

. SOLUTION. Step 3 . Add the results obtained in Step 2..

SOLUTION. Step 4 . Substitute the values into the formula. µ = ?[X·P(X)] µ = 4.30 Therefore, the mean or the expected value of the probability distribution is 4.30.

. Variance and Standard Deviation of a Discrete Random Variable.

. The variance, 02, ofa discrete random variable X is the number.

. Steps in Finding the Variance and Standard Deviation.

EXAMPLE. Find the variance and standard deviation of the probability distribution of the random variable X, which take only the values 1, 2, and 3, given that P(1)=10/33, P(2)=1/3, and P(3)=12/33..

. SOLUTION. Step 1. Find the mean of the probability distribution..

. SOLUTION. Step 2. Subtract the mean from each value of the random variable X.

. SOLUTION. Step 3. Square the results obtained in Step 2.

. SOLUTION. Step 4. Multiply the results obtained in Step 3 by the corresponding probability..

. Variance and Standard Deviation of a Discrete Random Variable.

. SOLUTION. Step 6. Get the square root of the variance to get the standard deviation.

END.