Shopping Outlet

A shopping outlet estimates the
probability distribution of the number of stores shoppers actually enter as shown in the table below.

x 0 1 2 3 4
p(x) 0.05 0.35 0.25 0.20 0.15

What is the probability that shoppers are just window shopping and not entering any stores?

What is the probability that shoppers enter at least 2 shops?

Find the expected value of the number of stores entered.

Find the variance and standard deviation of the number of stores entered.

Suppose that Y=2X+1. Use the laws of expected value to calculate the mean of Y from the probability distribution of X.

1. The probability that shoppers are just window shopping and not entering any stores is 0.05.

2. The probability that shoppers enter at least 2 shops is 0.75.
3. The expected value of the number of stores entered is 1.8.
4. The variance and standard deviation of the number of stores entered is 0.9 and 0.95, respectively.
5. The mean of Y from the probability distribution of X is 3.6.

To find the probability that shoppers are just window shopping and not entering any stores, we need to find the probability when x = 0.

The probability of shoppers entering 0 stores is given by p(x = 0), which is 0.05. So, the probability of shoppers just window shopping is 0.05.

To find the probability that shoppers enter at least 2 shops, we need to find the sum of probabilities when x is 2, 3, and 4.

The probability of shoppers entering at least 2 shops is given by:
p(x = 2) + p(x = 3) + p(x = 4)
= 0.25 + 0.20 + 0.15
= 0.60

So, the probability that shoppers enter at least 2 shops is 0.60.

To find the expected value of the number of stores entered, we multiply each possible value of x by its corresponding probability and sum them up.

Expected value (E) = (0 * 0.05) + (1 * 0.35) + (2 * 0.25) + (3 * 0.20) + (4 * 0.15)
= 0 + 0.35 + 0.50 + 0.60 + 0.60
= 2.05

So, the expected value of the number of stores entered is 2.05.

The variance of a random variable is the average squared deviation from the expected value. The standard deviation is the square root of the variance.

Variance (σ²) = (0 - 2.05)² * 0.05 + (1 - 2.05)² * 0.35 + (2 - 2.05)² * 0.25 + (3 - 2.05)² * 0.20 + (4 - 2.05)² * 0.15
= 1.95675

Standard deviation (σ) = √Variance = √1.95675 ≈ 1.4

So, the variance of the number of stores entered is 1.95675 and the standard deviation is approximately 1.4.

Given that Y = 2X + 1, we can use the linearity property of expected value to find the mean of Y.

E(Y) = E(2X + 1)
= 2E(X) + 1

Since we already calculated the expected value of X as 2.05, we can substitute that value into the equation:

E(Y) = 2 * 2.05 + 1
= 4.1 + 1
= 5.1

So, the mean of Y is 5.1.

To answer these questions, we need to use the given probability distribution. Let's break down the steps for each question:

1. Probability that shoppers are not entering any stores (window shopping):
The probability of shoppers not entering any stores is represented by p(0). Looking at the table, we see that p(0) = 0.05. Therefore, the probability that shoppers are just window shopping is 0.05.

2. Probability that shoppers enter at least 2 shops:
To find this probability, we need to calculate the sum of the probabilities for entering 2, 3, and 4 stores. Looking at the table, we see that p(2) = 0.25, p(3) = 0.20, and p(4) = 0.15. Adding them up, we get 0.25 + 0.20 + 0.15 = 0.60. Therefore, the probability that shoppers enter at least 2 shops is 0.60.

3. Expected value of the number of stores entered:
The expected value of a random variable can be calculated by multiplying each value by its corresponding probability and then summing them up. In this case, we multiply each value (0, 1, 2, 3, 4) by its respective probability (0.05, 0.35, 0.25, 0.20, 0.15) and sum them up.

Expected value = (0 * 0.05) + (1 * 0.35) + (2 * 0.25) + (3 * 0.20) + (4 * 0.15) = 0 + 0.35 + 0.50 + 0.60 + 0.60 = 2.05

Therefore, the expected value of the number of stores entered is 2.05.

4. Variance and standard deviation of the number of stores entered:
To calculate the variance, we need to find the squared difference between each value and the expected value, multiply it by its corresponding probability, and then sum them up. The formula for variance is Var(X) = Σ[(x - μ)^2 * P(x)], where x is the value, μ is the expected value, and P(x) is the probability for that value.

Variance = [(0 - 2.05)^2 * 0.05] + [(1 - 2.05)^2 * 0.35] + [(2 - 2.05)^2 * 0.25] + [(3 - 2.05)^2 * 0.20] + [(4 - 2.05)^2 * 0.15]

Calculating this expression, we get the variance as 1.6075.

The standard deviation, which measures the dispersion of the probability distribution, is simply the square root of the variance. So, taking the square root of the variance, we find that the standard deviation is approximately 1.27.

5. Mean of Y, when Y = 2X + 1:
If Y = 2X + 1, we can use the laws of expected value to calculate the mean of Y from the probability distribution of X. The general formula for this is E(Y) = E(2X + 1) = 2 * E(X) + 1.

Since we already calculated the expected value of X to be 2.05, we can plug it into the formula:

E(Y) = 2 * 2.05 + 1 = 4.10 + 1 = 5.10

Therefore, the mean of Y, when Y = 2X + 1, is 5.10.