The director of admissions at Kinzua University in Nova...

The director of admissions at Kinzua University in Nova Scotia estimated the distribution of student admissions for the fall semester on the basis of past experience. What is the expected number of admissions for the fall semester? Compute the variance and the standard deviation of the number of admissions. $$\begin{array}{|cc|}\hline \text { Admissions } & \text { Probability } \\\hline 1,000 & .6 \\1,200 & .3 \\ 1,500 & .1 \\\hline\end{array}$$

The director of admissions at Kinzua University in Nova Scotia estimated the distribution of student admissions for the fall semester on the basis of past experience. What is the expected number of admissions for the fall semester? Compute the variance and the standard deviation of the number of admissions.
$$\begin{array}{|cc|}\hline \text { Admissions } & \text { Probability } \\\hline 1,000 & .6 \\1,200 & .3 \\ 1,500 & .1 \\\hline\end{array}$$

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Key Concepts

Discrete Random Variable

A discrete random variable is one that takes on a countable number of distinct values, each associated with a probability. It is used to model outcomes of experiments where you can list all possible results, such as the number of admissions in a semester.

Probability Distribution

A probability distribution for a discrete random variable assigns a probability to each possible outcome. It ensures that the sum of these probabilities is 1 and provides the complete description of the behavior of the random variable.

Expected Value

The expected value, or mean, of a discrete random variable is calculated by taking the sum of each outcome multiplied by its corresponding probability. It provides a measure of the central tendency, representing the long-run average outcome if the experiment is repeated many times.

Variance

Variance measures the spread of a probability distribution by calculating the average of the squared differences between each outcome and the expected value. It quantifies the variability in the random variable’s outcomes.

Standard Deviation

The standard deviation is the square root of the variance, providing a measure of spread in the same units as the original variable. It reflects how much the outcomes of a random variable typically deviate from the expected value.

Transcript

00:01 For this exercise, we have the estimated probabilities for admissions numbers for a given college or university in nova scotia.

00:11 And we are asked to calculate the expected number of admissions for the fall semester and compute the variance and the standard deviation on the number of admissions.

00:22 So this is a probability distribution.

00:25 Let's call this x just for simplicity.

00:31 And so the probability is the probability of x.

00:36 So what is the expected number of admissions? the expected value of x is given by this formula, where the summation is over all possible values of x.

01:05 So this will be 1 ,000 times that probability, which is 0 .6, plus 1 ,200 times 0 .3, plus 1 ,500 times 0 .1.

01:17 And that comes out to 1 ,110.

01:22 So the expected number of admissions is 1 ,110.

01:29 The variance is given by this formula.

01:43 And here we have mu.

01:44 We just calculated that, 1 ,110.

01:48 We will calculate this in excel.

01:51 In a phase manner, first we will calculate x minus mu for each pair.

01:59 Then we will find that squared, and then multiply by the probability of x.

02:04 And finally, we will sum them over all possible values of x.

02:10 So here is our probability distribution in excel...