The given problem involves finding the probability of a Poisson variate x being between 1.5 and 2.9, given that the standard deviation of x is 2. To solve this problem, we need to follow a step-by-step approach.

Poisson Distribution:
The Poisson distribution is a discrete probability distribution that represents the number of events occurring in a fixed interval of time or space. It is characterized by a single parameter λ (lambda), which represents the average rate of events per interval.

Step 1: Understanding the Parameters:
In this problem, the given information is that the standard deviation of the Poisson variate x is 2. The standard deviation (σ) of a Poisson distribution is equal to the square root of its mean (λ). So, we can find the mean (λ) as follows:
σ = √λ
2 = √λ
Squaring both sides, we get:
4 = λ

Therefore, the mean (λ) of the Poisson distribution is 4.

Step 2: Calculating the Probability:
To find the desired probability, we need to calculate the cumulative probability of x being less than or equal to 2.9 and subtract the cumulative probability of x being less than or equal to 1.5.

P(1.5 ≤ x ≤ 2.9) = P(x ≤ 2.9) - P(x ≤ 1.5)

Step 3: Using Cumulative Distribution Function (CDF):
The cumulative distribution function (CDF) of a Poisson distribution gives the probability that the random variable is less than or equal to a specific value. We can use this function to calculate the desired probabilities.

P(x ≤ 2.9) = CDF(2.9, λ)
P(x ≤ 1.5) = CDF(1.5, λ)

Step 4: Applying the CDF Formula:
The CDF of a Poisson distribution can be calculated using the following formula:

CDF(x, λ) = Σ(k=0 to x) ((e^-λ) * (λ^k) / k!)

where e is the base of the natural logarithm (approximately 2.71828).

Using this formula, we can calculate the probabilities as follows:

P(x ≤ 2.9) = CDF(2.9, 4)
P(x ≤ 1.5) = CDF(1.5, 4)

Step 5: Substitute the Values and Calculate:
Substituting the values into the CDF formula and evaluating the sums, we can calculate the probabilities.

P(x ≤ 2.9) = CDF(2.9, 4) = 0.929
P(x ≤ 1.5) = CDF(1.5, 4) = 0.647

Step 6: Calculate the Final Probability:
Finally, we can calculate the desired probability by subtracting P(x ≤ 1.5) from P(x ≤ 2.9).

P(1.5 ≤ x ≤ 2.9) = P(x ≤ 2.9) - P(x ≤ 1.5)
= 0.929 - 0.647
= 0.282

Therefore, the probability of the Poisson variate x being between 1.5 and 2.9 is approximately 0.282

Answer is 0.144