Stats homework question please help urgently

Question

Suppose you are playing a game. The random variable, Y, represents your net profit in dollars. The probability distribution of Y is as follows. (Round your answers to 4 decimal places, if needed.)

y	-3	-2	0	3	5
P(Y = y)	0.1	0.16	?	0.12	0.1

(a) What is the mean for net profit?

(b) What is the median for net profit?

(c) What is the standard deviation for net profit?

Answer 1

We know that the total probability must sum to 1. So, to find the missing probability for y = 0:

0.1 + 0.16 + P(0) + 0.12 + 0.1 = 1

P(0) = 1 - (0.1 + 0.16 + 0.12 + 0.1) = 1 - 0.48 = 0.52

So, P(Y = 0) = 0.52.

(a) What is the mean for net profit?

The mean (expected value E(Y)) is calculated by multiplying each y value by its probability and summing them up:

E(Y) = (-3)(0.1) + (-2)(0.16) + (0)(0.52) + (3)(0.12) + (5)(0.1)

E(Y) = -0.3 + (-0.32) + 0 + 0.36 + 0.5 = 0.24

So, the mean net profit is 0.24 dollars.

(b) What is the median for net profit?

The median is the value of Y where the cumulative probability reaches or exceeds 0.5. Let's calculate the cumulative probabilities:

1. P(Y = -3) = 0.1
2. P(Y ≤ -2) = 0.1 + 0.16 = 0.26
3. P(Y ≤ 0) = 0.26 + 0.52 = 0.78

Since the cumulative probability exceeds 0.5 at Y = 0, the median net profit is 0.

(c) What is the standard deviation for net profit?

The standard deviation measures how spread out the values are around the mean. First, we calculate the variance, which is the expected value of the squared deviations from the mean:

Variance = Σ [P(Y=y) × (y - E(Y))^2]

We already know E(Y) = 0.24. Now we calculate each squared deviation:

Variance = (0.1)(-3 - 0.24)^2 + (0.16)(-2 - 0.24)^2 + (0.52)(0 - 0.24)^2 + (0.12)(3 - 0.24)^2 + (0.1)(5 - 0.24)^2

Now calculating each term:

Variance = 0.1(10.4976) + 0.16(4.9984) + 0.52(0.0576) + 0.12(7.6176) + 0.1(22.4976)

Variance = 1.04976 + 0.799744 + 0.029952 + 0.914112 + 2.24976 = 5.043328

Finally, the standard deviation is the square root of the variance:

Standard deviation = √5.043328 ≈ 2.245

Answer 2

The probability must total to 1 (100%) so P(0) = 0.52.

The mean is the sum of the probability and associated value of Y so:

Mean = (-3)(0.1) + (-2)(0.16) + (0)(0.52) + (3)(0.12) + (5)(0.1) = 0.24

The median is the value where the cumulative from either side is 0.5:

So it is zero.

The standard deviation (σ) is:

So √[(-3 - 0.24)²•(0.1) + (-2 - 0.24)²•(0.16) + (0 - 0.24)²•(0.52) + (3 - 0.24)²•(0.12) + (5 - 0.24)²•(0.1)]

√(1.04976 + 0.802816 + 0,029952 + 0.914112 + 2.26576) = √5.0624 = 2.2500