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| Question | Answer |
| CRV X doesn't have discrete values, so they are defined with.... | A range of values for X |
| Example of CRV : | |
| What is a probability density function (pdf)? | A graph of f(x) |
| How to plot pdf: | Sub in x-values into function and plot coordinates |
| 1. 1.5∫1 (2x/3)dx = 0.417 2. 2∫1.8 (2x/3)dx = 0.253 (use 2 as > ) 3. 1.2∫1.1 (2x/3)dx = 0.0767 4. 0 (as just a line with infinite thickness) | |
| For any CRV, X, P(X=x) always equals.... | 0 |
| The total area under a pdf must be exactly..... | 1 (∞∫-∞ f(x)dx = 1) Probability must be >0 as cant be negative |
| A continuous function must be a continuous function across all.... | It's sections (can be pdf when not continuous function) |
| What is the mode of a CRV? | The x-value which gets the max value for f(x) within the range of the pdf |
| How do we find the mode of a CRV? | Differentiation (e.g. is f(x) = 6x^2 - 4x^3 - 2x + 1 , dy/dx = 12x - 12x^2 - 2 = 0 , x = 0.211 , x = 0.789) x = 0.789 is the maximum value so the mode is 0.789 (Use d^2y/dx^2 to find if max or min + explanation + check if in range + check endpoints (incase maximum) |
| The median, Q2, of a CRV, X , is the value that satisfies : | Q2∫-∞ f(x)dx = 1/2 OR ∞∫Q2 f(x)dx = 1/2 Therefore: P(X≤Q2) = P(X≥Q2) = 1/2 |
| The lower quartile, Q1, is the value that satisfies : | Q1∫-∞ f(x)dx = 1/4 OR ∞∫Q1 f(x)dx = 3/4 |
| The upper quartile, Q1, is the value that satisfies : | Q3∫-∞ f(x)dx = 3/4 OR ∞∫Q3 f(x)dx = 1/4 |
| The mean of a CRV is: | E(X) = ∫ xf(x) dx |
| DRV vs CRV calculations (note s.d = root(var(x)) for both) | |
| For continuous random variable, X, Var(ax+b) = .... | a^2 (var(x)) |
| For continuous random variable, X, E(ax+b)= .... | aE(X) + b |
| For continuous random variable, X, s.d(ax+b) = ..... | ax |
| Note if E(X) = ∫ x(2x^2 /15) dx , then.... | E(X^2) = ∫x^2 (2x^2 /15) dx (E(X))^2 = ( ∫ x(2x^2 /15) dx )^2 |
| For CRV's, DRV's and Poisson distributions, we can combine independent means and variances : | E(X) + E(Y) = E(X + Y) Var(X) + Var(Y) = Var(X+Y) |
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