Cumulative distribution function - MATLAB cdf (2024)

Compute the cdf values for a normal distribution by specifying the distribution name 'Normal' and the distribution parameters.

Define the input vector x to contain the values at which to calculate the cdf.

x = [-2,-1,0,1,2];

Compute the cdf values for the normal distribution with the mean $$\mu$$ equal to 1 and the standard deviation $$\sigma$$ equal to 5.

mu = 1;sigma = 5;y = cdf('Normal',x,mu,sigma)

y = 1×5 0.2743 0.3446 0.4207 0.5000 0.5793

Each value in y corresponds to a value in the input vector x. For example, at the value x equal to 1, the corresponding cdf value y is equal to 0.5000.

Create a normal distribution object and compute the cdf values of the normal distribution using the object.

Create a normal distribution object with the mean $$\mu$$ equal to 1 and the standard deviation $$\sigma$$ equal to 5.

mu = 1;sigma = 5;pd = makedist('Normal','mu',mu,'sigma',sigma);

Define the input vector x to contain the values at which to calculate the cdf.

x = [-2,-1,0,1,2];

Compute the cdf values for the normal distribution at the values in x.

y = cdf(pd,x)

y = 1×5 0.2743 0.3446 0.4207 0.5000 0.5793

Each value in y corresponds to a value in the input vector x. For example, at the value x equal to 1, the corresponding cdf value y is equal to 0.5000.

Create a Poisson distribution object with the rate parameter, $$\lambda$$, equal to 2.

lambda = 2;pd = makedist('Poisson','lambda',lambda);

Define the input vector x to contain the values at which to calculate the cdf.

x = [0,1,2,3,4];

Compute the cdf values for the Poisson distribution at the values in x.

y = cdf(pd,x)

y = 1×5 0.1353 0.4060 0.6767 0.8571 0.9473

Each value in y corresponds to a value in the input vector x. For example, at the value x equal to 3, the corresponding cdf value y is equal to 0.8571.

Alternatively, you can compute the same cdf values without creating a probability distribution object. Use the cdf function, and specify a Poisson distribution using the same value for the rate parameter, $$\lambda$$.

y2 = cdf('Poisson',x,lambda)

y2 = 1×5 0.1353 0.4060 0.6767 0.8571 0.9473

The cdf values are the same as those computed using the probability distribution object.

Create a standard normal distribution object.

pd = makedist('Normal')

pd = NormalDistribution Normal distribution mu = 0 sigma = 1

Specify the x values and compute the cdf.

x = -3:.1:3;p = cdf(pd,x);

Plot the cdf of the standard normal distribution.

plot(x,p)

Create three gamma distribution objects. The first uses the default parameter values. The second specifies a = 1 and b = 2. The third specifies a = 2 and b = 1.

pd_gamma = makedist('Gamma')

pd_gamma = GammaDistribution Gamma distribution a = 1 b = 1

pd_12 = makedist('Gamma','a',1,'b',2)

pd_12 = GammaDistribution Gamma distribution a = 1 b = 2

pd_21 = makedist('Gamma','a',2,'b',1)

pd_21 = GammaDistribution Gamma distribution a = 2 b = 1

Specify the x values and compute the cdf for each distribution.

x = 0:.1:5;cdf_gamma = cdf(pd_gamma,x);cdf_12 = cdf(pd_12,x);cdf_21 = cdf(pd_21,x);

Create a plot to visualize how the cdf of the gamma distribution changes when you specify different values for the shape parameters a and b.

figure;J = plot(x,cdf_gamma);hold on;K = plot(x,cdf_12,'r--');L = plot(x,cdf_21,'k-.');set(J,'LineWidth',2);set(K,'LineWidth',2);legend([J K L],'a = 1, b = 1','a = 1, b = 2','a = 2, b = 1','Location','southeast');hold off;

Fit Pareto tails to a $$t$$ distribution at cumulative probabilities 0.1 and 0.9.

t = trnd(3,100,1);obj = paretotails(t,0.1,0.9);[p,q] = boundary(obj)

p = 2×1 0.1000 0.9000

q = 2×1 -1.8487 2.0766

Compute the cdf at the values in q.

cdf(obj,q)

ans = 2×1 0.1000 0.9000

y — cdf values
scalar value | array of scalar values

cdf values, returned as a scalar value or an array of scalar values. y is the same size as x after any necessary scalar expansion. Each element in y is the cdf value of the distribution, specified by the corresponding elements in the distribution parameters (A, B, C, and D) or the probability distribution object (pd), evaluated at the corresponding element in x.